Content check — Algebra and Trig, Chapter 10: Further Applications of Trigonometry

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Chapter 10: Further Applications of Trigonometry

376 problems · 155 figures

Try It #1 §10.1 source ↗

Solve the triangle shown in Figure 8 to the nearest tenth.

An oblique triangle with standard labels. Angle alpha is 98 degrees, angle gamma is 43 degrees, and side b is of length 22. Side b is the horizontal base.

Figure 8

Answer

\begin{array}{l} α = 98^{\circ} a = 34.6 \\ β = 39^{\circ} b = 22 \\ γ = 43^{\circ} c = 23.8 \end{array}

Try It #2 §10.1 source ↗

Given

α =

80°,

a =

120,

and

b = 121,

find the missing side and angles. If there is more than one possible solution, show both.

Answer

Solution 1

\begin{array}{l} α = 80° & a = 120 \\ β \approx 83.2° & b = 121 \\ γ \approx 16.8° & c \approx 35.2 \end{array}

Solution 2

\begin{array}{l} α^{'} = 80° a^{'} = 120 \\ β^{'} \approx 96.8° b^{'} = 121 \\ γ^{'} \approx 3.2° c^{'} \approx 6.8 \end{array}

Try It #3 §10.1 source ↗

Given

α =

80°,

a =

100,

b =

10,

find the missing side and angles. If there is more than one possible solution, show both. Round your answers to the nearest tenth.

Answer

β \approx

5.7°,

γ \approx

94.3°,

c \approx

101.3

Try It #4 §10.1 source ↗

Determine the number of triangles possible given

a = 31,

b = 26,

β = 48° .

Answer

two

Try It #5 §10.1 source ↗

Find the area of the triangle given

β = 42°,

a = 7.2 ft,

c = 3.4 ft .

Round the area to the nearest tenth.

Answer

about

8.2

square feet

Try It #6 §10.1 source ↗

The diagram shown in Figure 17 represents the height of a blimp flying over a football stadium. Find the height of the blimp if the angle of elevation at the southern end zone, point A, is 70°, the angle of elevation from the northern end zone, point $B,$ is 62°, and the distance between the viewing points of the two end zones is 145 yards.

An oblique triangle formed from three vertices A, B, and C. Verticies A and B are points on the ground, and vertex C is the blimp in the air between them. The distance between A and B is 145 yards. The angle at vertex A is 70 degrees, and the angle at vertex B is 62 degrees.

Figure 17

Answer

161.9 yd.

10.1-1 §10.1 source ↗

Describe the altitude of a triangle.

Answer

The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.

10.1-3 §10.1 source ↗

When can you use the Law of Sines to find a missing angle?

Answer

When the known values are the side opposite the missing angle and another side and its opposite angle.

10.1-5 §10.1 source ↗

What type of triangle results in an ambiguous case?

Answer

A triangle with two given sides and a non-included angle.

10.1-7 §10.1 source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$α =$ $35°,$ $γ =$ $73°,$ $c =$ $20$

Answer

β =

72°,

a \approx

12.0,

b \approx

19.9

10.1-9 §10.1 source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$a = 4,$ $α = 60°,$ $β = 100°$

Answer

γ =

20°,

b \approx

4.5,

c \approx

1.6

10.1-11 §10.1 source ↗

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $A$ is opposite side $a,$ angle $B$ is opposite side $b,$ and angle $C$ is opposite side $c .$

Find side $b$ when $A = 37°,$ $B = 49°,$ $c = 5.$

Answer

b \approx 3.78

10.1-13 §10.1 source ↗

Find side $c$ when $B =$ $37°,$ $C =$ $21°,$ $b = 23.$

Answer

c \approx 13.70

10.1-15 §10.1 source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

$γ =$ $113°,$ $b =$ $10,$ $c =$ $32$

Answer

one triangle,

α \approx

50.3°,

β \approx

16.7°,

a \approx

26.7

10.1-17 §10.1 source ↗

$a = 12,$ $c = 17,$ $α = 35°$

Answer

two triangles,

γ \approx

54.3°,

β \approx

90.7°,

b \approx

20.9

γ^{'} \approx

125.7°,

β^{'} \approx

19.3°,

b^{'} \approx

6.9

10.1-19 §10.1 source ↗

$a = 7,$ $c = 9,$ $α = 43°$

Answer

two triangles,

β \approx

75.7°,

γ \approx

61.3°,

b \approx

9.9

β^{'} \approx

18.3°,

γ^{'} \approx

118.7°,

b^{'} \approx

3.2

10.1-21 §10.1 source ↗

$b =$ $13,$ $c =$ $5,$ $γ =$ $10°$

Answer

two triangles,

α \approx

143.2°,

β \approx

26.8°,

a \approx

17.3

α^{'} \approx

16.8°,

β^{'} \approx

153.2°,

a^{'} \approx

8.3

10.1-23 §10.1 source ↗

$β =$ $119°,$ $b =$ $8.2,$ $a =$ $11.3$

Answer

no triangle possible

10.1-25 §10.1 source ↗

For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.

Find angle $A$ when $a =$ $13,$ $b =$ $6,$ $B =$ $20°.$

Answer

A \approx 47.8°

A^{'} \approx 132.2°

10.1-27 §10.1 source ↗

For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

$a =$ $5,$ $c =$ $6,$ $β =$ $35°$

Answer

8.6

10.1-29 §10.1 source ↗

For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

$a =$ $32,$ $b =$ $24,$ $γ =$ $75°$

Answer

370.9

10.1-31 §10.1 source ↗

For the following exercises, find the length of side $x .$ Round to the nearest tenth.

A triangle with an angle of 50 degrees and opposite side of length 10. Another angle is 70 degrees with side opposite of length x.

Answer

12.3

10.1-33 §10.1 source ↗

For the following exercises, find the length of side $x .$ Round to the nearest tenth.

A triangle. One angle is 45 degrees with side opposite = x. Another angle is 75 degrees. The side adjacent to the 45 and 75 degree angles = 15.

Answer

12.2

10.1-35 §10.1 source ↗

For the following exercises, find the length of side $x .$ Round to the nearest tenth.

A triangle. One angle is 50 degrees with opposite = x. Another angle is 42 degrees with opposite side = 14.

Answer

16.0

10.1-37 §10.1 source ↗

For the following exercises, find the measure of angle $x,$ if possible. Round to the nearest tenth.

A triangle. One angles is 98 degrees with opposite side = 10. Another angle is x degrees with opposite side = 5.

Answer

29.7°

10.1-39 §10.1 source ↗

For the following exercises, find the measure of angle $x,$ if possible. Round to the nearest tenth.

Answer

x =

76.9° or x =

103.1°

10.1-41 §10.1 source ↗

For the following exercises, find the measure of angle $x,$ if possible. Round to the nearest tenth.

Notice that $x$ is an obtuse angle.

A triangle. One angle is 55 degrees with side opposite = 21. Another angle is x degrees with opposite side = 24.

Answer

110.6°

10.1-43 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A triangle. One angle is 93 degrees with opposite side = 32.6. Another side is 24.1.

Answer

A \approx

39.4,

C \approx

47.6,

B C \approx

20.7

10.1-45 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A triangle. One angle is 25 degrees. The two sides adjacent to that angle are 18 and 15

Answer

57.1

10.1-47 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A triangle. One angle is 58 degrees with opposite side unknown. Another angle is 51 degrees with opposite side = 9. The side adjacent to the two given angles is 11.

Answer

42.0

10.1-49 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A triangle. One angle is 115 degrees with opposite side = 50. Another angle is 30 degrees with opposite side = 30.

Answer

430.2

10.1-51 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

Find the diameter of the circle in Figure 19. Round to the nearest tenth.

A triangle inscribed in a circle. Two of the legs are radii. The central angle formed by the radii is 110 degrees, and the opposite side is 8.3.

Figure 19

Answer

10.1

10.1-53 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

Find $A D$ in Figure 21. Round to the nearest tenth.

A triangle inside a triangle. The outer triangle is formed by vertices A, B, and D. Side B D is the base. The inner triangle shares vertices A and B. The last vertex C is located on the base side of the outer triangle between vertices B and D. Angle B is 53 degrees, angle D is 44 degrees, side A B is 12, and side A C is 13.

Figure 21

Answer

A D \approx 13.8

10.1-55 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

Find $A B$ in the parallelogram shown in Figure 23.

A parallelogram with vertices A, B, C, and D. There is a diagonal from vertex B to vertex C. Angle A is 130 degrees, angle D is 130 degrees, side B D is 10, and the diagonal B C is 12.

Figure 23

Answer

A B \approx 2.8

10.1-57 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

Solve the triangle in Figure 25. (Hint: Draw a perpendicular from $N$ to $L M).$ Round each answer to the nearest tenth.

A triangle with vertices M, N, and L. Side M N is the horizontal base and is 4.6. Angle M is 74 degrees, and side M L is 5.

Figure 25

Answer

L \approx

49.7,

N \approx

56.3,

L N \approx

5.8

10.1-59 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A pole leans away from the sun at an angle of $7°$ to the vertical, as shown in Figure 27. When the elevation of the sun is $55°,$ the pole casts a shadow 42 feet long on the level ground. How long is the pole? Round the answer to the nearest tenth.

A triangle within a triangle. The outer triangle is formed by vertices A, B, and S (the sun). Side A B is the horizontal base, the ground, and is 42 feet. Angle A is 55 degrees. The inner triangle is formed by vertices A, B, and C. Side B C is the pole. Vertex C is located on side A S of the outer triangle between vertices A and S. Angle C B S is 7 degrees.

Figure 27

Answer

51.4 feet

10.1-61 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

Figure 29 shows a satellite orbiting Earth. The satellite passes directly over two tracking stations $A$ and $B,$ which are 69 miles apart. When the satellite is on one side of the two stations, the angles of elevation at $A$ and $B$ are measured to be $83.9°$ and $86.2°,$ respectively. How far is the satellite from station $A$ and how high is the satellite above the ground? Round answers to the nearest whole mile.

A triangle formed by two ground tracking stations A and B and the satellite. Side A B is the horizontal base of the triangle. Angle A is 83.9 degrees, and the supplementary angle to angle B is 86.2 degrees.

Figure 29

Answer

The distance from the satellite to station

A

is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.

10.1-63 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

The roof of a house is at a $20°$ angle. An 8-foot solar panel is to be mounted on the roof and should be angled $38°$ relative to the horizontal for optimal results. (See Figure 31). How long does the vertical support holding up the back of the panel need to be? Round to the nearest tenth.

A triangle whose sides are the solar panel, the roof which goes past the solar panel, and the vertical support for the panel. The solar panel side is 8 feet long. There are horizontal dotted lines at the bottom of the solar panel and the bottom of the roof. The angle between the solar panel and the horizontal is 38 degrees. The angle between the roof and the horizontal is 20 degrees.

Figure 31

Answer

2.6 ft

10.1-65 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 4.3 km apart, to be 32° and 56°, as shown in Figure 33. Find the distance of the plane from point $A$ to the nearest tenth of a kilometer.

A triangle formed between the plane and two points on the ground, A and B. Side A B is the horizontal base. The plane is above and to the left of both A and B. Point B is to the right of point A. There is a dotted horizontal line going through the plane parallel to the ground. The angle formed between point B, the plane, and the dotted horizontal line is 32 degrees. The angle formed between point A, the plane, and the dotted horizontal line is 56 degrees.

Figure 33

Answer

5.6 km

10.1-67 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 35°. They then move 250 feet closer to the building and find the angle of elevation to be 53°. Assuming that the street is level, estimate the height of the building to the nearest foot.

Answer

371 ft

10.1-69 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A man and a woman standing $3 \frac{1}{2}$ miles apart spot a hot air balloon at the same time. If the angle of elevation from the man to the balloon is 27°, and the angle of elevation from the woman to the balloon is 41°, find the altitude of the balloon to the nearest foot.

Answer

5936 ft

10.1-71 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

A street light is mounted on a pole. A 6-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top of his head of 28°. A 6-foot-tall woman is standing on the same street on the opposite side of the pole from the man. The angle of elevation from the tip of her shadow to the top of her head is 28°. If the man and woman are 20 feet apart, how far is the street light from the tip of the shadow of each person? Round the distance to the nearest tenth of a foot.

Answer

24.1 ft

10.1-73 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

Two streets meet at an 80° angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 215 feet.

Answer

19,056 ft²

10.1-75 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

The Bermuda triangle is a region of the Atlantic Ocean that connects Bermuda, Florida, and Puerto Rico. Find the area of the Bermuda triangle if the distance from Florida to Bermuda is 1030 miles, the distance from Puerto Rico to Bermuda is 980 miles, and the angle created by the two distances is 62°.

Answer

445,624 square miles

10.1-77 §10.1 source ↗

For the following exercise, solve the triangle. Round each answer to the nearest tenth.

Naomi bought a dining table whose top is in the shape of a triangle. Find the area of the table top if two of the sides measure 4 feet and 4.5 feet, and the smaller angles measure 32° and 42°, as shown in Figure 35.

Figure 35

Answer

8.65 ft²

Try It #1 §10.2 source ↗

Find the missing side and angles of the given triangle:

α = 30°,

b = 12,

c = 24.

Answer

a \approx 14.9,

β \approx 23.8°,

γ \approx 126.2° .

Try It #2 §10.2 source ↗

Given

a = 5, b = 7,

and

c = 10,

find the missing angles.

Answer

α \approx 27.7°,

β \approx 40.5°,

γ \approx 111.8°

Try It #3 §10.2 source ↗

Use Heron’s formula to find the area of a triangle with sides of lengths

a =

29.7 ft,

b =

42.3 ft,

and

c = 38.4 ft .

Answer

Area = 552 square feet

Try It #4 §10.2 source ↗

Find the area of a triangle given

a =

4.38 ft,

b =

3.79 ft,

and

c = 5.22 ft .

Answer

about 8.15 square feet

10.2-1 §10.2 source ↗

If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?

Answer

two sides and the angle opposite the missing side.

10.2-3 §10.2 source ↗

Explain what

s

represents in Heron’s formula.

Answer

s

is the semi-perimeter, which is half the perimeter of the triangle.

10.2-5 §10.2 source ↗

When must you use the Law of Cosines instead of the Pythagorean Theorem?

Answer

The Law of Cosines must be used for any oblique (non-right) triangle.

10.2-7 §10.2 source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ If possible, solve each triangle for the unknown side. Round to the nearest tenth.

$α =$ $120°,$ $b =$ $6,$ $c =$ $7$

Answer

11.3

10.2-9 §10.2 source ↗

$γ =$ $115°,$ $a =$ $18,$ $b =$ $23$

Answer

34.7

10.2-11 §10.2 source ↗

$γ =$ $113°,$ $b =$ $10,$ $c =$ $32$

Answer

26.7

10.2-13 §10.2 source ↗

$α =$ $43.1°,$ $a =$ $184.2,$ $b =$ $242.8$

Answer

c =

257.3,

96.7

10.2-15 §10.2 source ↗

$β =$ $50°,$ $a =$ $105,$ $b =$ $45$

Answer

not possible

10.2-17 §10.2 source ↗

For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

$a =$ $14,$ $b =$ $13,$ $c =$ $20;$ find angle $C .$

Answer

95.5°

10.2-19 §10.2 source ↗

For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

$a =$ $13,$ $b =$ $22,$ $c =$ $28;$ find angle $A .$

Answer

26.9°

10.2-21 §10.2 source ↗

For the following exercises, solve the triangle. Round to the nearest tenth.

$A =$ $35°,$ $b =$ $8,$ $c =$ $11$

Answer

B \approx

45.9°,

C \approx

99.1°,

a \approx

6.4

10.2-23 §10.2 source ↗

For the following exercises, solve the triangle. Round to the nearest tenth.

$C =$ $121°,$ $a =$ $21,$ $b =$ $37$

Answer

A \approx

20.6°,

B \approx

38.4°,

c \approx

51.1

10.2-25 §10.2 source ↗

For the following exercises, solve the triangle. Round to the nearest tenth.

$a =$ $3.1,$ $b =$ $3.5,$ $c =$ $5$

Answer

A \approx

37.8°,

B \approx

43.8,

C \approx

98.4°

10.2-27 §10.2 source ↗

For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

Answer

177.56 in²

10.2-29 §10.2 source ↗

For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

$a =$ $\frac{1}{2} m,$ $b =$ $\frac{1}{3} m,$ $c =$ $\frac{1}{4} m$

Answer

0.04 m²

10.2-31 §10.2 source ↗

For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

$a =$ $1.6 yd,$ $b =$ $2.6 yd,$ $c =$ $4.1 yd$

Answer

0.91 yd²

10.2-33 §10.2 source ↗

For the following exercises, find the length of side $x .$ Round to the nearest tenth.

A triangle. One angle is 42 degrees with opposite side = x. The other two sides are 4.5 and 3.4.

Answer

3.0

10.2-35 §10.2 source ↗

For the following exercises, find the length of side $x .$ Round to the nearest tenth.

A triangle. One angle is 65 degrees with opposite side = x. The other two sides are 30 and 23.

Answer

29.1

10.2-37 §10.2 source ↗

For the following exercises, find the length of side $x .$ Round to the nearest tenth.

A triangle. One angle is 123 degrees with opposite side = x. The other two sides are 1/5 and 1/3.

Answer

0.5

10.2-39 §10.2 source ↗

For the following exercises, find the measurement of angle $A .$

A triangle. Angle A is opposite a side of length 125. The other two sides are 115 and 100.

Answer

70.7°

10.2-41 §10.2 source ↗

For the following exercises, find the measurement of angle $A .$

A triangle. Angle A is opposite a side of length 40.6. The other two sides are 38.7 and 23.3.

Answer

77.4°

10.2-43 §10.2 source ↗

For the following exercises, solve for the unknown side. Round to the nearest tenth.

A triangle. One angle is 60 degrees with opposite side unknown. The other two sides are 20 and 28.

Answer

25.0

10.2-45 §10.2 source ↗

For the following exercises, solve for the unknown side. Round to the nearest tenth.

A triangle. One angle is 22 degrees with opposite side unknown. The other two sides are 20 and 13.

Answer

9.3

10.2-47 §10.2 source ↗

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

A triangle with sides 8, 12, and 17. Angles unknown.

Answer

43.52

10.2-49 §10.2 source ↗

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

A triangle with sides 1.9, 2.6, and 4.3. Angles unknown.

Answer

1.41

10.2-51 §10.2 source ↗

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

A triangle with sides 1/2, 2/3, and 3/5. Angles unknown.

Answer

0.14

10.2-53 §10.2 source ↗

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 feet. Find the length of the shorter diagonal.

Answer

18.3

10.2-55 §10.2 source ↗

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

A regular octagon is inscribed in a circle with a radius of 8 inches. (See Figure 12.) Find the perimeter of the octagon.

Figure 12

Answer

48.98

10.2-57 §10.2 source ↗

For the following exercises, suppose that $x^{2} =$ $25 +$ $36 -$ $60 \cos (52)$ represents the relationship of three sides of a triangle and the cosine of an angle.

Draw the triangle.

Answer

A triangle. One angle is 52 degrees with opposite side = x. The other two sides are 5 and 6.

10.2-59 §10.2 source ↗

For the following exercises, find the area of the triangle.

Answer

7.62

10.2-61 §10.2 source ↗

For the following exercises, find the area of the triangle.

A triangle. One angle is 18 degrees with opposite side = 12.8. Another side is 18.8.

Answer

85.1

10.2-63 §10.2 source ↗

For the following exercises, find the area of the triangle.

A satellite calculates the distances and angle shown in Figure 15 (not to scale). Find the distance between the two cities. Round answers to the nearest tenth.

Insert figure(table) alt text: A triangle formed by two cities on the ground and a satellite above them. The angle by the satellite is 2.1 degrees with opposite side unknown, which is the distance between the two cities. The lengths of the other sides are 370 and 350 km.

Figure 15

Answer

24.0 km

10.2-65 §10.2 source ↗

For the following exercises, find the area of the triangle.

A 113-foot tower is located on a hill that is inclined 34° to the horizontal, as shown in Figure 16. A guy-wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed.

Insert figure(table) alt text: Two triangles, one on top of the other. The bottom triangle is the hill inclined 34 degrees to the horizontal. The second is formed by the base of the tower on the incline of the hill, the top of the tower, and the wire anchor point uphill from the tower on the incline. The sides are the tower, the incline of the hill, and the wire. The tower side is 113 feet and the incline side is 98 feet.

Figure 16

Answer

99.9 ft

10.2-67 §10.2 source ↗

For the following exercises, find the area of the triangle.

The graph in Figure 17 represents two boats departing at the same time from the same dock. The first boat is traveling at 18 miles per hour at a heading of 327° and the second boat is traveling at 4 miles per hour at a heading of 60°. Find the distance between the two boats after 2 hours.

Insert figure(table) alt text: A graph of two rays, which represent the paths of the two boats. Both rays start at the origin. The first goes into the first quadrant at a 60 degree angle at 4 mph. The second goes into the fourth quadrant at a 327 degree angle from the origin. The second travels at 18 mph.

Figure 17

Answer

37.3 miles

10.2-69 §10.2 source ↗

For the following exercises, find the area of the triangle.

A pilot flies in a straight path for 1 hour 30 min. She then makes a course correction, heading 10° to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?

Answer

2371 miles

10.2-71 §10.2 source ↗

For the following exercises, find the area of the triangle.

Philadelphia is 140 miles from Washington, D.C., Washington, D.C. is 442 miles from Boston, and Boston is 315 miles from Philadelphia. Draw a triangle connecting these three cities and find the angles in the triangle.

Answer

10.2-73 §10.2 source ↗

For the following exercises, find the area of the triangle.

Two airplanes take off in different directions. One travels 300 mph due west and the other travels 25° north of west at 420 mph. After 90 minutes, how far apart are they, assuming they are flying at the same altitude?

Answer

292.4 miles

10.2-75 §10.2 source ↗

For the following exercises, find the area of the triangle.

The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is 117°. What is the area of this quadrilateral?

Answer

65.4 cm²

10.2-77 §10.2 source ↗

For the following exercises, find the area of the triangle.

Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132°. Round to the nearest whole square foot.

Answer

468 ft²

Try It #1 §10.3 source ↗

Plot the point

(2, \frac{π}{3})

in the polar grid.

Answer

Polar grid with point (2, pi/3) plotted.

Try It #2 §10.3 source ↗

Plot the points

(3, - \frac{π}{6})

and

(2, \frac{9 π}{4})

on the same polar grid.

Answer

Points (2, 9pi/4) and (3, -pi/6) are plotted in the polar grid.

Try It #3 §10.3 source ↗

Write the polar coordinates

(- 1, \frac{2 π}{3})

as rectangular coordinates.

Answer

(x, y) =

(\frac{1}{2}, - \frac{\sqrt{3}}{2})

Try It #4 §10.3 source ↗

Rewrite the Cartesian equation

y^{2} = 3 - x^{2}

in polar form.

Answer

r = \sqrt{3}

Try It #5 §10.3 source ↗

Rewrite the polar equation

r = 2 \sin θ

in Cartesian form.

Answer

x^{2} + y^{2} = 2 y

or, in the standard form for a circle,

x^{2} + {(y - 1)}^{2} = 1

10.3-1 §10.3 source ↗

How are polar coordinates different from rectangular coordinates?

Answer

For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.

10.3-3 §10.3 source ↗

Explain how polar coordinates are graphed.

Answer

Determine

θ

for the point, then move

r

units from the pole to plot the point. If

r

is negative, move

r

units from the pole in the opposite direction but along the same angle. The point is a distance of

r

away from the origin at an angle of

θ

from the polar axis.

10.3-5 §10.3 source ↗

Explain why the points

(- 3, \frac{π}{2})

and

(3, - \frac{π}{2})

are the same.

Answer

The point

(- 3, \frac{π}{2})

has a positive angle but a negative radius and is plotted by moving to an angle of

\frac{π}{2}

and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point

(3, - \frac{π}{2})

has a negative angle and a positive radius and is plotted by first moving to an angle of

- \frac{π}{2}

and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.

10.3-7 §10.3 source ↗

For the following exercises, convert the given polar coordinates to Cartesian coordinates. Remember to consider the quadrant in which the given point is located when determining $θ$ for the point.

$(5, π)$

Answer

(- 5, 0)

10.3-9 §10.3 source ↗

For the following exercises, convert the given polar coordinates to Cartesian coordinates. Remember to consider the quadrant in which the given point is located when determining $θ$ for the point.

$(- 3, \frac{π}{6})$

Answer

(- \frac{3 \sqrt{3}}{2}, - \frac{3}{2})

10.3-11 §10.3 source ↗

For the following exercises, convert the given Cartesian coordinates to polar coordinates with $r >$ $0,$ $0 \leq θ <$ $2 π .$ Remember to consider the quadrant in which the given point is located.

$(4, 2)$

Answer

(2 \sqrt{5}, 0.464)

10.3-13 §10.3 source ↗

For the following exercises, convert the given Cartesian coordinates to polar coordinates with $r >$ $0,$ $0 \leq θ <$ $2 π .$ Remember to consider the quadrant in which the given point is located.

$(3, −5)$

Answer

(\sqrt{34}, 5.253)

10.3-15 §10.3 source ↗

For the following exercises, convert the given Cartesian coordinates to polar coordinates with $r >$ $0,$ $0 \leq θ <$ $2 π .$ Remember to consider the quadrant in which the given point is located.

$(8, 8)$

Answer

(8 \sqrt{2}, \frac{π}{4})

10.3-17 §10.3 source ↗

For the following exercises, convert the given Cartesian equation to a polar equation.

$y = 4$

Answer

r = 4 \csc θ

10.3-19 §10.3 source ↗

For the following exercises, convert the given Cartesian equation to a polar equation.

$y = 2 x^{4}$

Answer

r =

\sqrt[3]{\frac{s i n θ}{2 c o s^{4} θ}}

10.3-21 §10.3 source ↗

For the following exercises, convert the given Cartesian equation to a polar equation.

$x^{2} + y^{2} = 3 x$

Answer

r = 3 \cos θ

10.3-23 §10.3 source ↗

For the following exercises, convert the given Cartesian equation to a polar equation.

$x^{2} - y^{2} = 3 y$

Answer

r =

\frac{3 \sin θ}{\cos (2 θ)}

10.3-25 §10.3 source ↗

For the following exercises, convert the given Cartesian equation to a polar equation.

$x^{2} = 9 y$

Answer

r =

\frac{9 \sin θ}{\cos^{2} θ}

10.3-27 §10.3 source ↗

For the following exercises, convert the given Cartesian equation to a polar equation.

$9 x y = 1$

Answer

r =

\sqrt{\frac{1}{9 \cos θ \sin θ}}

10.3-29 §10.3 source ↗

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r = 4 \cos θ$

Answer

x^{2} + y^{2} = 4 x

\frac{{(x - 2)}^{2}}{4} +

\frac{y^{2}}{4} =

1;

circle

10.3-31 §10.3 source ↗

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r =$ $\frac{6}{\cos θ + 3 \sin θ}$

Answer

3 y + x = 6;

line

10.3-33 §10.3 source ↗

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r = 3 \csc θ$

Answer

y = 3;

line

10.3-35 §10.3 source ↗

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r^{2} =$ $4 \sec θ \csc θ$

Answer

x y = 4;

hyperbola

10.3-37 §10.3 source ↗

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r^{2} = 4$

Answer

x^{2} + y^{2} = 4;

circle

10.3-39 §10.3 source ↗

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r =$ $\frac{3}{\cos θ - 5 \sin θ}$

Answer

x - 5 y = 3;

line

10.3-41 §10.3 source ↗

For the following exercises, find the polar coordinates of the point.

Polar coordinate system with a point located on the third concentric circle and midway between pi/2 and pi in the second quadrant.

Answer

(3, \frac{3 π}{4})

10.3-43 §10.3 source ↗

For the following exercises, find the polar coordinates of the point.

Polar coordinate system with a point located on the fifth concentric circle and pi.

Answer

(5, π)

10.3-45 §10.3 source ↗

For the following exercises, plot the points.

$(- 2, \frac{π}{3})$

Answer

Polar coordinate system with a point located on the second concentric circle and two-thirds of the way between pi and 3pi/2 (closer to 3pi/2).

10.3-47 §10.3 source ↗

For the following exercises, plot the points.

$(3.5, \frac{7 π}{4})$

Answer

Polar coordinate system with a point located midway between the third and fourth concentric circles and midway between 3pi/2 and 2pi.

10.3-49 §10.3 source ↗

For the following exercises, plot the points.

$(5, \frac{π}{2})$

Answer

Polar coordinate system with a point located on the fifth concentric circle and pi/2.

10.3-51 §10.3 source ↗

For the following exercises, plot the points.

$(3, \frac{5 π}{6})$

Answer

Polar coordinate system with a point located on the third concentric circle and 2/3 of the way between pi/2 and pi (closer to pi).

10.3-53 §10.3 source ↗

For the following exercises, plot the points.

$(- 2, \frac{π}{4})$

Answer

Polar coordinate system with a point located on the second concentric circle and midway between pi and 3pi/2.

10.3-55 §10.3 source ↗

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

$5 x - y = 6$

Answer

$r =$ $\frac{6}{5 \cos θ - \sin θ}$

Plot of given line in the polar coordinate grid

10.3-57 §10.3 source ↗

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

$x^{2} + {(y - 1)}^{2} = 1$

Answer

$r = 2 \sin θ$

Plot of given circle in the polar coordinate grid

10.3-59 §10.3 source ↗

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

$x = 2$

Answer

$r = \frac{2}{\cos θ}$

10.3-61 §10.3 source ↗

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

$x^{2} + y^{2} = 3 x$

Answer

$r = 3 \cos θ$

10.3-63 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

$r = - 4$

Answer

$x^{2} + y^{2} = 16$

Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.

10.3-65 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

$θ = \frac{π}{4}$

Answer

$y = x$

Plot of line y=x in the rectangular coordinates grid.

10.3-67 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

$r = −10 \sin θ$

Answer

$x^{2} +$ ${(y + 5)}^{2} =$ $25$

Plot of circle with radius 5 centered at (0,-5).

10.3-69 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

Use a graphing calculator to find the rectangular coordinates of $(2, - \frac{π}{5}) .$ Round to the nearest thousandth.

Answer

(1.618, - 1.176)

10.3-71 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

Use a graphing calculator to find the polar coordinates of $(- 7, 8)$ in degrees. Round to the nearest thousandth.

Answer

(10.630, 131.186°)

10.3-73 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

Use a graphing calculator to find the polar coordinates of $(- 2, 0)$ in radians. Round to the nearest hundredth.

Answer

(2, 3.14) o r (2, π)

10.3-75 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

Describe the graph of $r =$ $a \sec θ;$ $a <$ $0.$

Answer

A vertical line with

a

units left of the y-axis.

10.3-77 §10.3 source ↗

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

Describe the graph of $r =$ $a \csc θ;$ $a <$ $0.$

Answer

A horizontal line with

a

units below the x-axis.

10.3-79 §10.3 source ↗

For the following exercise, graph the polar inequality.

$r < 4$

Answer

Graph of shaded circle of radius 4 with the edge not included (dotted line) - polar coordinate grid.

10.3-81 §10.3 source ↗

For the following exercise, graph the polar inequality.

$θ = \frac{π}{4}, r \geq 2$

Answer

Graph of ray starting at (2, pi/4) and extending in a positive direction along pi/4 - polar coordinate grid.

10.3-83 §10.3 source ↗

For the following exercise, graph the polar inequality.

$0 \leq θ \leq \frac{π}{3},$ $r <$ $2$

Answer

Graph of the shaded region 0 to pi/3 from r=0 to 2 with the edge not included (dotted line) - polar coordinate grid

Try It #1 §10.4 source ↗

Test the equation for symmetry:

r = - 2 \cos θ .

Answer

The equation fails the symmetry test with respect to the line

θ = \frac{π}{2}

and with respect to the pole. It passes the polar axis symmetry test.

Try It #2 §10.4 source ↗

Without converting to Cartesian coordinates, test the given equation for symmetry and find the zeros and maximum values of

| r | :

r = 3 \cos θ .

Answer

Tests will reveal symmetry about the polar axis. The zero is

(0, \frac{π}{2}),

and the maximum value is

(3, 0) .

Try It #3 §10.4 source ↗

Sketch the graph of

r = 3 - 2 \cos θ .

Answer

Graph of the limaçon r=3-2cos(theta). Extending to the left.

Try It #4 §10.4 source ↗

Sketch the graph of

r = 4 \sin (2 θ) .

Answer

The graph is a rose curve, $n$ even

Graph of rose curve r=4 sin(2 theta). Even - four petals equally spaced, each of length 4.

Try It #5 §10.4 source ↗

Sketch the graph of

r = 3 \cos (3 θ).

Answer

Graph of rose curve r=3cos(3theta). Three petals equally spaced from origin.

Rose curve, $n$ odd

Try It #6 §10.4 source ↗

Sketch the graph of

r = - θ

over the interval

[0, 4 π] .

Answer

A dark blue spiral curve is plotted on a polar coordinate grid with concentric circles and radial lines. The curve starts near the origin and expands outwards, resembling an Archimedean spiral. The grid includes labeled axes from -15 to 15, with concentric circles indicating radii at intervals of 5, up to a maximum radius of 15.

10.4-1 §10.4 source ↗

Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.

Answer

Symmetry with respect to the polar axis is similar to symmetry about the

x

-axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line

θ = \frac{π}{2}

is similar to symmetry about the

y

-axis.

10.4-3 §10.4 source ↗

What are the steps to follow when graphing polar equations?

Answer

Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at

θ = 0, \frac{π}{2},

π

and

\frac{3 π}{2},

and sketch the graph.

10.4-5 §10.4 source ↗

What part of the equation determines the shape of the graph of a polar equation?

Answer

The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.

10.4-7 §10.4 source ↗

For the following exercises, test the equation for symmetry.

$r = 3 - 3 \cos θ$

Answer

symmetric with respect to the polar axis

10.4-9 §10.4 source ↗

For the following exercises, test the equation for symmetry.

$r = 3 \sin 2 θ$

Answer

symmetric with respect to the polar axis, symmetric with respect to the line

θ = \frac{π}{2},

symmetric with respect to the pole

10.4-11 §10.4 source ↗

For the following exercises, test the equation for symmetry.

$r = 2 θ$

Answer

symmetric with respect to the line

θ = \frac{π}{2}

10.4-13 §10.4 source ↗

For the following exercises, test the equation for symmetry.

$r = \frac{2}{θ}$

Answer

Symmetric with respect to line

θ = \frac{π}{2}

(y-axis)

10.4-15 §10.4 source ↗

For the following exercises, test the equation for symmetry.

$r = \sqrt{5 \sin 2 θ}$

Answer

symmetric with respect to the pole

10.4-17 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 4 \sin θ$

Answer

circle

10.4-19 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 2 - 2 \cos θ$

Answer

cardioid

10.4-21 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 3 + 3 \sin θ$

Answer

cardioid

10.4-23 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 7 + 4 \sin θ$

Answer

one-loop/dimpled limaçon

10.4-25 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 5 + 4 \cos θ$

Answer

one-loop/dimpled limaçon

10.4-27 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 1 + 3 \sin θ$

Answer

inner loop/two-loop limaçon

10.4-29 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 5 + 7 \sin θ$

Answer

inner loop/two-loop limaçon

10.4-31 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 5 + 6 \cos θ$

Answer

inner loop/two-loop limaçon

10.4-33 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r^{2} =$ $10 \cos (2 θ)$

Answer

lemniscate

10.4-35 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r^{2} =$ $10 \sin (2 θ)$

Answer

lemniscate

10.4-37 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 3 cos (2 θ)$

Answer

rose curve

10.4-39 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = 4 sin (4 θ)$

Answer

rose curve

10.4-41 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = - θ$

Answer

Archimedes’ spiral

10.4-43 §10.4 source ↗

For the following exercises, graph the polar equation. Identify the name of the shape.

$r = - 3 θ$

Answer

Archimedes’ spiral

10.4-45 §10.4 source ↗

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

$r = \frac{1}{\sqrt{θ}}$

Answer

10.4-47 §10.4 source ↗

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

$r = 2 \sqrt{1 - \sin^{2} θ}$ , a hippopede

Answer

Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin)

10.4-49 §10.4 source ↗

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

$r = 2 - \sin (2 θ)$

Answer

10.4-51 §10.4 source ↗

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

$r = θ + 1$

Answer

Graph of given equation. Similar to original Archimedes' spiral.

10.4-53 §10.4 source ↗

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

$r = θ \cos θ$

Answer

10.4-55 §10.4 source ↗

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of $[0, 4 π]$ and then explain the differences shown in the graphs.

$r =$ $θ,$ $r =$ $θ +$ $\sin θ$

Answer

They are both spirals, but not quite the same.

10.4-57 §10.4 source ↗

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of $[0, 4 π]$ and then explain the differences shown in the graphs.

$r =$ $2 \sin (\frac{θ}{2}),$ $r =$ $θ \sin (\frac{θ}{2})$

Answer

Both graphs are curves with 2 loops. The equation with a coefficient of

θ

has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to

4 π

to get a better picture.

10.4-59 §10.4 source ↗

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of $[0, 4 π]$ and then explain the differences shown in the graphs.

On a graphing utility, graph $r = \sin (\frac{16}{5} θ)$ on $[0$ , $4 π]$ , $[0$ , $8 π]$ , $[0$ , $12 π]$ , and $[0$ , $16 π] .$ Describe the effect of increasing the width of the domain.

Answer

When the width of the domain is increased, more petals of the flower are visible.

10.4-61 §10.4 source ↗

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of $[0, 4 π]$ and then explain the differences shown in the graphs.

On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

\begin{array}{l} r_{1} = 3 \sin (3 θ) \\ r_{2} = 2 \sin (3 θ) \\ r_{3} = \sin (3 θ) \end{array}

Answer

The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.

10.4-63 §10.4 source ↗

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of $[0, 4 π]$ and then explain the differences shown in the graphs.

On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

\begin{array}{l} r_{1} = 3 θ \\ r_{2} = 2 θ \\ r_{3} = θ \end{array}

Answer

The graphs are spirals. The smaller the coefficient, the tighter the spiral.

10.4-65 §10.4 source ↗

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1} =$ $6 -$ $4 \cos θ,$ $r_{2} =$ $4$

Answer

(4, \frac{π}{3}),

(4, \frac{5 π}{3})

10.4-67 §10.4 source ↗

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1} =$ $1 +$ $\cos θ,$ $r_{2} =$ $3 \cos θ$

Answer

(\frac{3}{2}, \frac{π}{3}),

(\frac{3}{2}, \frac{5 π}{3})

10.4-69 §10.4 source ↗

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1} = \sin^{2} (2 θ)$ , $r_{2} =$ $1 -$ $\cos (4 θ)$

Answer

(0, \frac{π}{2}),

(0, π),

(0, \frac{3 π}{2}),

(0, 2 π)

10.4-71 §10.4 source ↗

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.

$r_{1}^{2} =$ $\sin θ,$ $r_{2}^{2} =$ $\cos θ$

Answer

= \frac{π}{4}

and

\sqrt{sin \frac{π}{4}} =

\sqrt{\frac{\sqrt{2}}{2}} =

\frac{\sqrt[4]{8}}{2}

Try It #1 §10.5 source ↗

Plot the point

1 + 5 i

in the complex plane.

Answer

Try It #2 §10.5 source ↗

Find the absolute value of the complex number

z = 12 - 5 i .

Answer

Try It #3 §10.5 source ↗

Given

z = 1 - 7 i,

find

| z | .

Answer

| z | = \sqrt{50} = 5 \sqrt{2}

Try It #4 §10.5 source ↗

Express

z = 3 i

r cis θ

in polar form.

Answer

z =

3 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))

Try It #5 §10.5 source ↗

Write

z = \sqrt{3} + i

in polar form.

Answer

z =

2 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))

Try It #6 §10.5 source ↗

Convert the complex number to rectangular form:

z = 4 (\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})

Answer

z = 2 \sqrt{3} - 2 i

Try It #7 §10.5 source ↗

Find the product and the quotient of

z_{1} =

2 \sqrt{3} (\cos (150°) + i \sin (150°))

and

z_{2} =

2 (\cos (30°) + i \sin (30°)) .

Answer

z_{1} z_{2} =

-

4 \sqrt{3};

\frac{z_{1}}{z_{2}} =

-

\frac{\sqrt{3}}{2} +

\frac{3}{2} i

Try It #8 §10.5 source ↗

Find the four fourth roots of

16 (\cos (120°) + i \sin (120°)) .

Answer

$z_{0} =$ $2 (\cos (30°) + i \sin (30°))$

$z_{1} =$ $2 (\cos (120°) + i \sin (120°))$

$z_{2} =$ $2 (\cos (210°) + i \sin (210°))$

$z_{3} =$ $2 (\cos (300°) + i \sin (300°))$

10.5-1 §10.5 source ↗

A complex number is

a + b i .

Explain each part.

Answer

a is the real part, b is the imaginary part, and

i = \sqrt{- 1}

10.5-3 §10.5 source ↗

How is a complex number converted to polar form?

Answer

Polar form converts the real and imaginary part of the complex number in polar form using

x = r \cos θ

and

y = r \sin θ .

10.5-5 §10.5 source ↗

What is De Moivre’s Theorem and what is it used for?

Answer

z^{n} =

r^{n} (\cos (n θ) + i \sin (n θ))

It is used to simplify polar form when a number has been raised to a power.

10.5-7 §10.5 source ↗

For the following exercises, find the absolute value of the given complex number.

$- 7 + i$

Answer

5 \sqrt{2}

10.5-9 §10.5 source ↗

For the following exercises, find the absolute value of the given complex number.

$\sqrt{2} - 6 i$

Answer

\sqrt{38}

10.5-11 §10.5 source ↗

For the following exercises, find the absolute value of the given complex number.

$2.2 - 3.1 i$

Answer

\sqrt{14.45}

10.5-13 §10.5 source ↗

For the following exercises, write the complex number in polar form.

$8 - 4 i$

Answer

4 \sqrt{5} cis (333.4°)

10.5-15 §10.5 source ↗

For the following exercises, write the complex number in polar form.

$\sqrt{3} + i$

Answer

2 cis (\frac{π}{6})

10.5-17 §10.5 source ↗

For the following exercises, convert the complex number from polar to rectangular form.

$z = 7 cis (\frac{π}{6})$

Answer

\frac{7 \sqrt{3}}{2} + i \frac{7}{2}

10.5-19 §10.5 source ↗

For the following exercises, convert the complex number from polar to rectangular form.

$z = 4 cis (\frac{7 π}{6})$

Answer

- 2 \sqrt{3} - 2 i

10.5-21 §10.5 source ↗

For the following exercises, convert the complex number from polar to rectangular form.

$z =$ $3 cis (240°)$

Answer

- 1.5 - i \frac{3 \sqrt{3}}{2}

10.5-23 §10.5 source ↗

For the following exercises, find $z_{1} z_{2}$ in polar form.

$z_{1} =$ $2 \sqrt{3} cis (116°);$ $z_{2} =$ $2 cis (82°)$

Answer

4 \sqrt{3} cis (198°)

10.5-25 §10.5 source ↗

For the following exercises, find $z_{1} z_{2}$ in polar form.

$z_{1} =$ $3 cis (120°);$ $z_{2} =$ $\frac{1}{4} cis (60°)$

Answer

\frac{3}{4} cis (180°)

10.5-27 §10.5 source ↗

For the following exercises, find $z_{1} z_{2}$ in polar form.

$z_{1} =$ $\sqrt{5} cis (\frac{5 π}{8});$ $z_{2} =$ $\sqrt{15} cis (\frac{π}{12})$

Answer

5 \sqrt{3} cis (\frac{17 π}{24})

10.5-29 §10.5 source ↗

For the following exercises, find $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} =$ $21 cis (135°);$ $z_{2} =$ $3 cis (65°)$

Answer

7 cis (70°)

10.5-31 §10.5 source ↗

For the following exercises, find $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} =$ $15 cis (120°);$ $z_{2} =$ $3 cis (40°)$

Answer

5 cis (80°)

10.5-33 §10.5 source ↗

For the following exercises, find $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} =$ $5 \sqrt{2} cis (π);$ $z_{2} =$ $\sqrt{2} cis (\frac{2 π}{3})$

Answer

5 cis (\frac{π}{3})

10.5-35 §10.5 source ↗

For the following exercises, find the powers of each complex number in polar form.

Find $z^{3}$ when $z =$ $5 cis (45°) .$

Answer

125 cis (135°)

10.5-37 §10.5 source ↗

For the following exercises, find the powers of each complex number in polar form.

Find $z^{2}$ when $z =$ $3 cis (120°) .$

Answer

9 cis (240°)

10.5-39 §10.5 source ↗

For the following exercises, find the powers of each complex number in polar form.

Find $z^{4}$ when $z =$ $cis (\frac{3 π}{16}) .$

Answer

cis (\frac{3 π}{4})

10.5-41 §10.5 source ↗

For the following exercises, evaluate each root.

Evaluate the cube root of $z$ when $z =$ $27 cis (240°) .$

Answer

3 cis (80°),

3 cis (200°),

3 cis (320°)

10.5-43 §10.5 source ↗

For the following exercises, evaluate each root.

Evaluate the cube root of $z$ when $z =$ $32 cis (\frac{2 π}{3}) .$

Answer

2 \sqrt[3]{4} cis (\frac{2 π}{9}),

2 \sqrt[3]{4} cis (\frac{8 π}{9}),

2 \sqrt[3]{4} cis (\frac{14 π}{9})

10.5-45 §10.5 source ↗

For the following exercises, evaluate each root.

Evaluate the square root of $z$ when $z =$ $8 cis (\frac{7 π}{4}) .$

Answer

2 \sqrt{2} cis (\frac{7 π}{8}),

2 \sqrt{2} cis (\frac{15 π}{8})

10.5-47 §10.5 source ↗

For the following exercises, plot the complex number in the complex plane.

$- 3 - 3 i$

Answer

Plot of -3 -3i in the complex plane (-3 along real axis, -3 along imaginary axis).

10.5-49 §10.5 source ↗

For the following exercises, plot the complex number in the complex plane.

$- 1 - 5 i$

Answer

Plot of -1 -5i in the complex plane (-1 along real axis, -5 along imaginary axis).

10.5-51 §10.5 source ↗

For the following exercises, plot the complex number in the complex plane.

$2 i$

Answer

Plot of 2i in the complex plane (0 along the real axis, 2 along the imaginary axis).

10.5-53 §10.5 source ↗

For the following exercises, plot the complex number in the complex plane.

$6 - 2 i$

Answer

Plot of 6-2i in the complex plane (6 along the real axis, -2 along the imaginary axis).

10.5-55 §10.5 source ↗

For the following exercises, plot the complex number in the complex plane.

$1 - 4 i$

Answer

Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).

10.5-57 §10.5 source ↗

For the following exercises, find all answers rounded to the nearest hundredth.

Use the rectangular to polar feature on the graphing calculator to change $3 - 2 i$ to polar form.

Answer

3.61 e^{- 0.59 i}

10.5-59 §10.5 source ↗

For the following exercises, find all answers rounded to the nearest hundredth.

Use the polar to rectangular feature on the graphing calculator to change $4 cis (120°)$ to rectangular form.

Answer

- 2 + 3.46 i

10.5-61 §10.5 source ↗

For the following exercises, find all answers rounded to the nearest hundredth.

Use the polar to rectangular feature on the graphing calculator to change $5 cis (210°)$ to rectangular form.

Answer

- 4.33 - 2.50 i

Try It #1 §10.6 source ↗

Construct a table of values and plot the parametric equations:

x (t) = t - 3,

y (t) =

2 t +

4;

-

1 \leq t \leq 2.

Answer

$t$	$x (t)$	$y (t)$
$- 1$	$- 4$	$2$
$0$	$- 3$	$4$
$1$	$- 2$	$6$
$2$	$- 1$	$8$

A coordinate plane displays a blue line segment ascending from (-4, 2) to (0, 10). Arrows indicate its direction along grid lines, highlighting its positive slope.

Try It #2 §10.6 source ↗

Parameterize the curve given by

x = y^{3} - 2 y .

Answer

\begin{array}{l} x (t) = t^{3} - 2 t \\ y (t) = t \end{array}

Try It #3 §10.6 source ↗

Given the equations below, eliminate the parameter and write as a rectangular equation for $y$ as a function of $x .$

\begin{array}{l} x (t) = 2 t^{2} + 6 \\ y (t) = 5 - t \end{array}

Answer

y = 5 - \sqrt{\frac{1}{2} x - 3}

Try It #4 §10.6 source ↗

Eliminate the parameter and write as a rectangular equation.

\begin{array}{l} x (t) = t^{2} \\ y (t) = \ln t t > 0 \end{array}

Answer

y = \ln \sqrt{x}

Try It #5 §10.6 source ↗

Eliminate the parameter from the given pair of parametric equations and write as a Cartesian equation:

x (t) = 2 \cos t

and

y (t) = 3 \sin t .

Answer

\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1

Try It #6 §10.6 source ↗

Write the given parametric equations as a Cartesian equation:

x (t) = t^{3}

and

y (t) = t^{6} .

Answer

y = x^{2}

10.6-1 §10.6 source ↗

What is a system of parametric equations?

Answer

A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example,

x = f (t)

and

y = f (t) .

10.6-3 §10.6 source ↗

Explain how to eliminate a parameter given a set of parametric equations.

Answer

Choose one equation to solve for

t,

substitute into the other equation and simplify.

10.6-5 §10.6 source ↗

What is a benefit of using parametric equations?

Answer

Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.

10.6-7 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{array}{l} x (t) = 5 - t \\ y (t) = 8 - 2 t \end{array}$

Answer

y = - 2 + 2 x

10.6-9 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{array}{l} x (t) = 2 t + 1 \\ y (t) = 3 \sqrt{t} \end{array}$

Answer

y = 3 \sqrt{\frac{x - 1}{2}}

10.6-11 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{array}{l} x (t) = 2 e^{t} \\ y (t) = 1 - 5 t \end{array}$

Answer

x = 2 e^{\frac{1 - y}{5}}

y = 1 - 5 l n (\frac{x}{2})

10.6-13 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{cases} x (t) = 4 log (t) \\ y (t) = 3 + 2 t \end{cases}$

Answer

x =

4 \log (\frac{y - 3}{2})

10.6-15 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{array}{l} x (t) = t^{3} - t \\ y (t) = 2 t \end{array}$

Answer

x = {(\frac{y}{2})}^{3} - \frac{y}{2}

10.6-17 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{array}{l} x (t) = e^{2 t} \\ y (t) = e^{6 t} \end{array}$

Answer

y = x^{3}

10.6-19 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{cases} x (t) = 4 cos t \\ y (t) = 5 \sin t \end{cases}$

Answer

{(\frac{x}{4})}^{2} +

{(\frac{y}{5})}^{2} =

1

10.6-21 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{cases} x (t) = 2 {cos}^{2} t \\ y (t) = - \sin t \end{cases}$

Answer

y^{2} = 1 - \frac{1}{2} x

10.6-23 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{cases} x (t) = t - 1 \\ y (t) = t^{2} \end{cases}$

Answer

y = x^{2} + 2 x + 1

10.6-25 §10.6 source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{cases} x (t) = 2 t - 1 \\ y (t) = t^{3} - 2 \end{cases}$

Answer

y = {(\frac{x + 1}{2})}^{3} - 2

10.6-27 §10.6 source ↗

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an $x - y$ table.

${\begin{cases} x (t) = 4 - t \\ y (t) = 3 t + 2 \end{cases}$

Answer

y = - 3 x + 14

10.6-29 §10.6 source ↗

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an $x - y$ table.

${\begin{cases} x (t) = 4 t - 1 \\ y (t) = 4 t + 2 \end{cases}$

Answer

y = x + 3

10.6-31 §10.6 source ↗

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting $x (t) = t$ or by setting $y (t) = t .$

$y (x) =$ $2 \sin x +$ $1$

Answer

{\begin{array}{l} x (t) = t \\ y (t) = 2 \sin t + 1 \end{array}

10.6-33 §10.6 source ↗

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting $x (t) = t$ or by setting $y (t) = t .$

$x (y) = \sqrt{y} + 2 y$

Answer

{\begin{array}{l} x (t) = \sqrt{t} + 2 t \\ y (t) = t \end{array}

10.6-35 §10.6 source ↗

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x (t) = a \cos t$ and $y (t) = b \sin t .$ Identify the curve.

$\frac{x^{2}}{16} + \frac{y^{2}}{36} = 1$

Answer

{\begin{array}{l} x (t) = 4 \cos t \\ y (t) = 6 \sin t \end{array};

Ellipse

10.6-37 §10.6 source ↗

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x (t) = a \cos t$ and $y (t) = b \sin t .$ Identify the curve.

$x^{2} + y^{2} = 10$

Answer

{\begin{array}{l} x (t) = \sqrt{10} \cos t \\ y (t) = \sqrt{10} \sin t \end{array};

Circle

10.6-39 §10.6 source ↗

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x (t) = a \cos t$ and $y (t) = b \sin t .$ Identify the curve.

Parameterize the line from $(−1, 0)$ to $(3, −2)$ so that the line is at $(−1, 0)$ at $t = 0,$ and at $(3, −2)$ at $t = 1.$

Answer

{\begin{array}{l} x (t) = - 1 + 4 t \\ y (t) = - 2 t \end{array}

10.6-41 §10.6 source ↗

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x (t) = a \cos t$ and $y (t) = b \sin t .$ Identify the curve.

Parameterize the line from $(4, 1)$ to $(6, −2)$ so that the line is at $(4, 1)$ at $t = 0,$ and at $(6, −2)$ at $t = 1.$

Answer

{\begin{array}{l} x (t) = 4 + 2 t \\ y (t) = 1 - 3 t \end{array}

10.6-43 §10.6 source ↗

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.

${\begin{array}{l} x_{1} (t) = t^{2} \\ y_{1} (t) = 2 t - 1 \end{array} and {\begin{array}{l} x_{2} (t) = - t + 6 \\ y_{2} (t) = t + 1 \end{array}$

Answer

yes, at

t = 2

10.6-45 §10.6 source ↗

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

${\begin{array}{l} x_{1} (t) = t^{2} - 4 \\ y_{1} (t) = 2 t^{2} - 1 \end{array}$

$t$	$x$	$y$
1
2
3

Answer

$t$	$x$	$y$
1	-3	1
2	0	7
3	5	17

10.6-47 §10.6 source ↗

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

Find two different sets of parametric equations for $y = {(x + 1)}^{2} .$

Answer

answers may vary:

{\begin{array}{l} x (t) = t - 1 \\ y (t) = t^{2} \end{array} and {\begin{array}{l} x (t) = t + 1 \\ y (t) = {(t + 2)}^{2} \end{array}

10.6-49 §10.6 source ↗

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

Find two different sets of parametric equations for $y = x^{2} - 4 x + 4.$

Answer

answers may vary: ,

{\begin{array}{l} x (t) = t \\ y (t) = t^{2} - 4 t + 4 \end{array} and {\begin{array}{l} x (t) = t + 2 \\ y (t) = t^{2} \end{array}

Try It #1 §10.7 source ↗

Sketch the graph of the parametric equations

x = \sqrt{t},

y = 2 t + 3,

0 \leq t \leq 3.

Answer

Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.

Try It #2 §10.7 source ↗

Graph the parametric equations:

x = 5 \cos t,

y = 3 \sin t .

Answer

Graph of the given equations - a horizontal ellipse.

Try It #3 §10.7 source ↗

Sketch the graph of the parametric equations

x =

2 \cos θ and y =

4 \sin θ,

along with the rectangular equation on the same grid.

Answer

The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.

Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.

10.7-1 §10.7 source ↗

What are two methods used to graph parametric equations?

Answer

plotting points with the orientation arrow and a graphing calculator

10.7-3 §10.7 source ↗

Why are some graphs drawn with arrows?

Answer

The arrows show the orientation, the direction of motion according to increasing values of

t .

10.7-5 §10.7 source ↗

Why are parametric graphs important in understanding projectile motion?

Answer

The parametric equations show the different vertical and horizontal motions over time.

10.7-7 §10.7 source ↗

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.

${\begin{array}{l} x (t) = t - 1 \\ y (t) = t^{2} \end{array}$

$t$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$
$x$
$y$

Answer

Graph of the given equations - looks like an upward opening parabola.

10.7-9 §10.7 source ↗

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.

${\begin{array}{l} x (t) = - 2 - 2 t \\ y (t) = 3 + t \end{array}$

$t$	$- 3$	$- 2$	$- 1$	$0$	$1$
$x$
$y$

Answer

Graph of the given equations - a line, negative slope.

10.7-11 §10.7 source ↗

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.

${\begin{array}{l} x (t) = t^{2} \\ y (t) = t + 3 \end{array}$

$t$	$- 2$	$- 1$	$0$	$1$	$2$
$x$
$y$

Answer

Graph of the given equations - looks like a sideways parabola, opening to the right.

10.7-13 §10.7 source ↗

For the following exercises, sketch the curve and include the orientation.

${\begin{cases} x (t) = - \sqrt{t} \\ y (t) = t \end{cases}$

Answer

Graph of the given equations - looks like the left half of an upward opening parabola.

10.7-15 §10.7 source ↗

For the following exercises, sketch the curve and include the orientation.

${\begin{cases} x (t) = - t + 2 \\ y (t) = 5 - | t | \end{cases}$

Answer

Graph of the given equations - looks like a downward opening absolute value function.

10.7-17 §10.7 source ↗

For the following exercises, sketch the curve and include the orientation.

${\begin{cases} x (t) = 2 sin t \\ y (t) = 4 cos t \end{cases}$

Answer

Graph of the given equations - a vertical ellipse.

10.7-19 §10.7 source ↗

For the following exercises, sketch the curve and include the orientation.

${\begin{cases} x (t) = 3 \cos^{2} t \\ y (t) = −3 \sin^{2} t \end{cases}$

Answer

Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.

10.7-21 §10.7 source ↗

For the following exercises, sketch the curve and include the orientation.

${\begin{cases} x (t) = \sec t \\ y (t) = \tan^{2} t \end{cases}$

Answer

10.7-23 §10.7 source ↗

For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.

${\begin{array}{l} x (t) = t - 1 \\ y (t) = - t^{2} \end{array}$

Answer

Graph of the given equations- looks like a downward opening parabola.

10.7-25 §10.7 source ↗

For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.

${\begin{cases} x (t) = 2 \cos t \\ y (t) = - \sin t \end{cases}$

Answer

Graph of the given equations- horizontal ellipse.

10.7-27 §10.7 source ↗

For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.

${\begin{cases} x (t) = e^{2 t} \\ y (t) = - e^{t} \end{cases}$

Answer

Graph of the given equations- looks like the lower half of a sideways parabola opening to the right

10.7-29 §10.7 source ↗

For the following exercises, graph the equation and include the orientation.

$x =$ $2 t,$ $y =$ $t^{2},$ $-$ $5 \leq t \leq 5$

Answer

Graph of the given equations- looks like an upwards opening parabola

10.7-31 §10.7 source ↗

For the following exercises, graph the equation and include the orientation.

$x (t) =$ $-$ $t,$ $y (t) =$ $\sqrt{t},$ $t \geq 0$

Answer

Graph of the given equations- looks like the upper half of a sideways parabola opening to the left

10.7-33 §10.7 source ↗

For the following exercises, graph the equation and include the orientation.

$x = - \sec t,$ $y = \tan t,$ $-$ $\frac{π}{2} <$ $t <$ $\frac{π}{2}$

Answer

Graph of the given equations- the left half of a hyperbola with diagonal asymptotes

10.7-35 §10.7 source ↗

For the following exercises, use the parametric equations for integers a and b:

\begin{array}{l} x (t) = a \cos ((a + b) t) \\ y (t) = a \cos ((a - b) t) \end{array}

Graph on the domain $[- π, 0],$ where $a = 3$ and $b = 2$ , and include the orientation.

Answer

Graph of the given equations - vertical periodic trajectory

10.7-37 §10.7 source ↗

For the following exercises, use the parametric equations for integers a and b:

\begin{array}{l} x (t) = a \cos ((a + b) t) \\ y (t) = a \cos ((a - b) t) \end{array}

Graph on the domain $[- π, 0],$ where $a = 5$ and $b = 4$ , and include the orientation.

Answer

10.7-39 §10.7 source ↗

For the following exercises, use the parametric equations for integers a and b:

\begin{array}{l} x (t) = a \cos ((a + b) t) \\ y (t) = a \cos ((a - b) t) \end{array}

Describe the graph if $a = 100$ and $b = 99.$

Answer

There will be 100 back-and-forth motions.

10.7-41 §10.7 source ↗

For the following exercises, use the parametric equations for integers a and b:

\begin{array}{l} x (t) = a \cos ((a + b) t) \\ y (t) = a \cos ((a - b) t) \end{array}

If the parametric equations $x (t) = t^{2}$ and $y (t) = 6 - 3 t$ have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?

Answer

Take the opposite of the

x (t)

equation.

10.7-43 §10.7 source ↗

For the following exercises, describe the graph of the set of parametric equations.

$y (t) = t^{2}$ and $x (t)$ is linear

Answer

The parabola opens up.

10.7-45 §10.7 source ↗

For the following exercises, describe the graph of the set of parametric equations.

Write the parametric equations of a circle with center $(0, 0),$ radius 5, and a counterclockwise orientation.

Answer

{\begin{cases} x (t) = 5 \cos t \\ y (t) = 5 \sin t \end{cases}

10.7-47 §10.7 source ↗

For the following exercises, use a graphing utility to graph on the window $[- 3, 3]$ by $[- 3, 3]$ on the domain $[0, 2 π)$ for the following values of $a$ and $b$ , and include the orientation.

{\begin{cases} x (t) = \sin (a t) \\ y (t) = \sin (b t) \end{cases}

$a = 1, b = 2$

Answer

10.7-49 §10.7 source ↗

For the following exercises, use a graphing utility to graph on the window $[- 3, 3]$ by $[- 3, 3]$ on the domain $[0, 2 π)$ for the following values of $a$ and $b$ , and include the orientation.

{\begin{cases} x (t) = \sin (a t) \\ y (t) = \sin (b t) \end{cases}

$a = 3, b = 3$

Answer

Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 1 unit.

10.7-51 §10.7 source ↗

For the following exercises, use a graphing utility to graph on the window $[- 3, 3]$ by $[- 3, 3]$ on the domain $[0, 2 π)$ for the following values of $a$ and $b$ , and include the orientation.

{\begin{cases} x (t) = \sin (a t) \\ y (t) = \sin (b t) \end{cases}

$a = 2, b = 5$

Answer

Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 3 units.

10.7-53 §10.7 source ↗

For the following exercises, look at the graphs that were created by parametric equations of the form ${\begin{cases} x (t) = a cos (b t) \\ y (t) = c sin (d t) \end{cases} .$ Use the parametric mode on the graphing calculator to find the values of $a, b, c,$ and $d$ to achieve each graph.

Answer

a = 4,

b = 3,

c = 6,

d = 1

10.7-55 §10.7 source ↗

Answer

a = 4,

b = 2,

c = 3,

d = 3

10.7-57 §10.7 source ↗

For the following exercises, use a graphing utility to graph the given parametric equations.

Graph all three sets of parametric equations on the domain $[0,$ $2 π] .$

Answer

10.7-59 §10.7 source ↗

For the following exercises, use a graphing utility to graph the given parametric equations.

Graph all three sets of parametric equations on the domain $[- 4 π, 6 π] .$

Answer

10.7-61 §10.7 source ↗

For the following exercises, use a graphing utility to graph the given parametric equations.

Explain the effect on the graph of the parametric equation when we switched $\sin t$ and $\cos t$ .

Answer

The

y

-intercept changes.

10.7-63 §10.7 source ↗

For the following exercises, use a graphing utility to graph the given parametric equations.

An object is thrown in the air with vertical velocity of 20 ft/s and horizontal velocity of 15 ft/s. The object’s height can be described by the equation $y (t) =$ $-$ $16 t^{2} +$ $20 t$ , while the object moves horizontally with constant velocity 15 ft/s. Write parametric equations for the object’s position, and then eliminate time to write height as a function of horizontal position.

Answer

y (x) =

-

16 {(\frac{x}{15})}^{2} +

20 (\frac{x}{15})

10.7-65 §10.7 source ↗

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time $t .$ Neglect air resistance.

Find parametric equations that model the problem situation.

Answer

{\begin{cases} x (t) = 64 t \cos (52 °) \\ y (t) = - 16 t^{2} + 64 t \sin (52 °) \end{cases}

10.7-67 §10.7 source ↗

When will the dart hit the ground?

Answer

approximately 3.2 seconds

10.7-69 §10.7 source ↗

At what time will the dart reach maximum height?

Answer

1.6 seconds

10.7-71 §10.7 source ↗

For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.

A hypocycloid: ${\begin{array}{l} x (t) = 6 \sin t + 2 \sin (6 t) \\ y (t) = 6 \cos t - 2 \cos (6 t) \end{array}$ on the domain $[0, 2 π]$ .

Answer

Graph of the given equations - a hypocycloid

10.7-73 §10.7 source ↗

A rose: ${\begin{array}{l} x (t) = 5 \sin (2 t) \sin t \\ y (t) = 5 \sin (2 t) \cos t \end{array}$ on the domain $[0, 2 π]$ .

Answer

Graph of the given equations - a four petal rose

Try It #1 §10.8 source ↗

Draw a vector $v$ that connects from the origin to the point

(3, 5) .

Answer

A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.

Try It #2 §10.8 source ↗

Find the scalar multiple 3 $u$ given $u$

= 〈 5, 4 〉 .

Answer

3 u = 〈 15, 12 〉

Try It #3 §10.8 source ↗

Write the vector $u$ with initial point

P = (- 1, 6)

and terminal point

Q = (7, - 5)

in terms of $i$ and $j .$

Answer

u = 8 i - 11 j

Try It #4 §10.8 source ↗

A vector travels from the origin to the point

(3, 5) .

Write the vector in terms of magnitude and direction.

Answer

$v =$ $\sqrt{34} \cos (59°) i +$ $\sqrt{34} \sin (59°) j$

Magnitude = $\sqrt{34}$

$θ =$ $\tan^{- 1} (\frac{5}{3}) =$ $59.04°$

10.8-1 §10.8 source ↗

What are the characteristics of the letters that are commonly used to represent vectors?

Answer

lowercase, bold letter, usually

u, v, w

10.8-3 §10.8 source ↗

What are $i$ and $j,$ and what do they represent?

Answer

They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.

10.8-5 §10.8 source ↗

When a unit vector is expressed as

〈 a, b 〉,

which letter is the coefficient of the $i$ and which the $j ?$

Answer

The first number always represents the coefficient of the

i,

and the second represents the

j .

10.8-7 §10.8 source ↗

Given a vector with initial point

(- 4, 2)

and terminal point

(3, - 3),

find an equivalent vector whose initial point is

(0, 0) .

Write the vector in component form

〈 a, b 〉 .

Answer

〈 7, - 5 〉

10.8-9 §10.8 source ↗

For the following exercises, determine whether the two vectors $u$ and $v$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2}$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4}$ .

$P_{1} =$ $(5, 1),$ $P_{2} =$ $(3, - 2),$ $P_{3} =$ $(- 1, 3),$ and $P_{4} = (9, - 4)$

Answer

not equal

10.8-11 §10.8 source ↗

$P_{1} =$ $(- 1, - 1),$ $P_{2} =$ $(- 4, 5),$ $P_{3} =$ $(- 10, 6),$ and $P_{4} = (- 13, 12)$

Answer

equal

10.8-13 §10.8 source ↗

$P_{1} =$ $(8, 3),$ $P_{2} =$ $(6, 5),$ $P_{3} =$ $(11, 8),$ and $P_{4} = (9, 10)$

Answer

equal

10.8-15 §10.8 source ↗

Given initial point $P_{1} = (6, 0)$ and terminal point $P_{2} = (- 1, - 3),$ write the vector $v$ in terms of $i$ and $j .$

Answer

- 7 i - 3 j

10.8-17 §10.8 source ↗

For the following exercises, use the vectors u = i + 5j, v = −2i− 3j, and w = 4i − j.

Find 4v + 2u

Answer

- 6 i - 2 j

10.8-19 §10.8 source ↗

For the following exercises, use the given vectors to compute u + v, u − v, and 2u − 3v.

$u =$ $〈 -$ $3,$ $4 〉,$ $v =$ $〈 -$ $2,$ $1 〉$

Answer

u +

v =

〈 -

5,

5 〉,

u -

v =

〈 -

1,

3 〉,

2 u -

3 v =

〈 0,

5 〉

10.8-21 §10.8 source ↗

For the following exercises, use the given vectors to compute u + v, u − v, and 2u − 3v.

Let v = 5i + 2j. Find a vector that is twice the length and points in the opposite direction as $v .$

Answer

- 10 i - 4 j

10.8-23 §10.8 source ↗

For the following exercises, find a unit vector in the same direction as the given vector.

b = −2i + 5j

Answer

-

\frac{2 \sqrt{29}}{29} i +

\frac{5 \sqrt{29}}{29} j

10.8-25 §10.8 source ↗

For the following exercises, find a unit vector in the same direction as the given vector.

$d = - \frac{1}{3} i + \frac{5}{2} j$

Answer

-

\frac{2 \sqrt{229}}{229} i +

\frac{15 \sqrt{229}}{229} j

10.8-27 §10.8 source ↗

For the following exercises, find a unit vector in the same direction as the given vector.

u = −14i + 2j

Answer

- \frac{7 \sqrt{2}}{10} i + \frac{\sqrt{2}}{10} j

10.8-29 §10.8 source ↗

For the following exercises, find the magnitude and direction of the vector, $0 \leq θ < 2 π .$

$〈 6, 5 〉$

Answer

| v | =

7.810,

θ =

39.806 °

10.8-31 §10.8 source ↗

For the following exercises, find the magnitude and direction of the vector, $0 \leq θ < 2 π .$

$〈 −4, −6 〉$

Answer

| v | =

7.211,

θ =

236.310°

10.8-33 §10.8 source ↗

For the following exercises, find the magnitude and direction of the vector, $0 \leq θ < 2 π .$

Given u = −i − j and v = i + 5j, calculate $u \cdot v .$

Answer

- 6

10.8-35 §10.8 source ↗

For the following exercises, find the magnitude and direction of the vector, $0 \leq θ < 2 π .$

Given u $= 〈 - 1, 6 〉$ and v $= 〈 6, - 1 〉,$ calculate $u \cdot v .$

Answer

- 12

10.8-37 §10.8 source ↗

For the following exercises, given $v,$ draw $v,$ 3v and $\frac{1}{2} v .$

$〈 −1, 4 〉$

Answer

Three graphs illustrate scalar multiplication of a vector. The first graph shows vector v. The second shows vector 3v, which is three times longer than v. The third shows vector 1/2v, half the length of v.

10.8-39 §10.8 source ↗

For the following exercises, use the vectors shown to sketch u + v, u − v, and 2u.

Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,1) and v goes to (-1,2).

Answer

Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).

10.8-41 §10.8 source ↗

For the following exercises, use the vectors shown to sketch u + v, u − v, and 2u.

Plot of vectors u and v located head to tail. Take u's start point as the origin. In terms of that, u goes from the origin to (3,-2), and v goes from (3,-2) to (2,-3)

Answer

Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).

10.8-43 §10.8 source ↗

For the following exercises, use the vectors shown to sketch 2u + v.

Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,-2) and v goes from the origin to (-3,-2).

Answer

Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).

10.8-45 §10.8 source ↗

For the following exercises, use the vectors shown to sketch u − 3v.

Answer

Vector extending from the origin to (7,5), taking the base as the origin.

10.8-47 §10.8 source ↗

For the following exercises, write the vector shown in component form.

Answer

〈 4, 1 〉

10.8-49 §10.8 source ↗

For the following exercises, write the vector shown in component form.

Given initial point $P_{1} = (4, - 1)$ and terminal point $P_{2} = (- 3, 2),$ write the vector $v$ in terms of $i$ and $j .$ Draw the points and the vector on the graph.

Answer

$v = - 7 i + 3 j$

10.8-51 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

$| v | = 6, θ = 45 °$

Answer

3 \sqrt{2} i + 3 \sqrt{2} j

10.8-53 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

$| v | =$ $2,$ $θ =$ $300°$

Answer

i - \sqrt{3} j

10.8-55 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

A 60-pound box is resting on a ramp that is inclined 12°. Rounding to the nearest tenth,

ⓐ Find the magnitude of the normal (perpendicular) component of the force.

ⓑ Find the magnitude of the component of the force that is parallel to the ramp.

Answer

ⓐ 58.7

ⓑ 12.5

10.8-57 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27° above the horizontal. Round to the nearest hundredth.

Answer

x = 7.13

pounds,

y = 3.63

pounds

10.8-59 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55° above the horizontal. Round to the nearest hundredth.

Answer

x = 2.87

pounds,

y = 4.10

pounds

10.8-61 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?

Answer

4.635 miles, 17.764° N of E

10.8-63 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?

Answer

17 miles. 10.318 miles

10.8-65 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

A man starts walking from home and walks 3 miles at 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how far would he have to the walk, and in what direction?

Answer

Distance: 2.868. Direction: 86.474° North of West, or 3.526° West of North

10.8-67 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

Answer

4.924°. 659 km/hr

10.8-69 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

Answer

4.424°

10.8-71 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

As part of a video game, the point $(5, 7)$ is rotated counterclockwise about the origin through an angle of 35°. Find the new coordinates of this point.

Answer

(0.081, 8.602)

10.8-73 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

Answer

21.801°, relative to the car’s forward direction

10.8-75 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

A 50-pound object rests on a ramp that is inclined 19°. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.

Answer

parallel: 16.28, perpendicular: 47.28 pounds

10.8-77 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it ─135° from the horizontal, and 5 pounds acting on it directed 150° from the horizontal. What single force is the resultant force acting on the body?

Answer

19.35 pounds, 231.54° from the horizontal

10.8-79 §10.8 source ↗

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

Answer

5.1583 pounds, 75.8° from the horizontal

10.R-1 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$β =$ $50°,$ $a =$ $105,$ $b =$ $45$

Answer

Not possible

10.R-3 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

Solve the triangle.

Triangle with standard labels. Angle A is 36 degrees with opposite side a unknown. Angle B is 24 degrees with opposite side b = 16. Angle C and side c are unknown.

Answer

C =

120°,

a =

23.1,

c =

34.1

10.R-5 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in Figure 1. Find the distance of the plane from point $A$ and the elevation of the plane.

Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.

Figure 1

Answer

distance of the plane from point

A :

2.2 km, elevation of the plane: 1.6 km

10.R-7 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

Solve the triangle in Figure 2, rounding to the nearest tenth.

A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13.

Figure 2

Answer

b =

71.0°,

C =

55.0°,

a =

12.8

10.R-9 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

To find the distance between two cities, a satellite calculates the distances and angle shown in Figure 3 (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees.

Figure 3

Answer

40.6 km

10.R-11 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

Plot the point with polar coordinates $(5, - \frac{2 π}{3})$

Answer

Polar coordinate grid with a point plotted on the fifth concentric circle 2/3 the way between pi and 3pi/2 (closer to 3pi/2).

10.R-13 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

Convert $(- 2, \frac{3 π}{2})$ to rectangular coordinates.

Answer

(0, 2)

10.R-15 §10.R source ↗

For the following exercises, assume $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

Convert $(- 9, - 4)$ to polar coordinates.

Answer

(9.8489, 203.96°)

10.R-17 §10.R source ↗

For the following exercises, convert the given Cartesian equation to a polar equation.

$x^{2} + y^{2} = 64$

Answer

r = 8

10.R-19 §10.R source ↗

For the following exercises, convert the given polar equation to a Cartesian equation.

$r = 7 cos θ$

Answer

x^{2} + y^{2} = 7 x

10.R-21 §10.R source ↗

For the following exercises, convert to rectangular form and graph.

$θ = \frac{3 π}{4}$

Answer

$y = - x$

Plot of the function y=-x in rectangular coordinates.

10.R-23 §10.R source ↗

For the following exercises, test each equation for symmetry.

$r = 4 + 4 \sin θ$

Answer

symmetric with respect to the line

θ = \frac{π}{2}

10.R-25 §10.R source ↗

For the following exercises, test each equation for symmetry.

Sketch a graph of the polar equation $r = 1 - 5 \sin θ .$ Label the axis intercepts.

Answer

Graph of the given polar equation - an inner loop limaçon.

10.R-27 §10.R source ↗

For the following exercises, test each equation for symmetry.

Sketch a graph of the polar equation $r = 3 - 3 \cos θ$

Answer

Graph of the given polar equation - a cardioid.

10.R-29 §10.R source ↗

For the following exercises, find the absolute value of each complex number.

$4 - 3 i$

Answer

10.R-31 §10.R source ↗

Write the complex number in polar form.

$\frac{1}{2} - \frac{\sqrt{3}}{2} i$

Answer

cis (- \frac{π}{3})

10.R-33 §10.R source ↗

For the following exercises, convert the complex number from polar to rectangular form.

$z = 3 cis (40°)$

Answer

2.3 + 1.9 i

10.R-35 §10.R source ↗

For the following exercises, find the product $z_{1} z_{2}$ in polar form.

$z_{1} =$ $10 cis (\frac{π}{6})$

$z_{2} = 6 cis (\frac{π}{3})$

Answer

60 cis (\frac{π}{2})

10.R-37 §10.R source ↗

For the following exercises, find the quotient $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} =$ $27 cis (\frac{5 π}{3})$

$z_{2} = 9 cis (\frac{π}{3})$

Answer

3 cis (\frac{4 π}{3})

10.R-39 §10.R source ↗

For the following exercises, find the powers of each complex number in polar form.

Find $z^{2}$ when $z = 5 cis (\frac{3 π}{4})$

Answer

25 cis (\frac{3 π}{2})

10.R-41 §10.R source ↗

For the following exercises, evaluate each root.

Evaluate the square root of $z$ when $z =$ $25 cis (\frac{3 π}{2}) .$

Answer

5 cis (\frac{3 π}{4}),

5 cis (\frac{7 π}{4})

10.R-43 §10.R source ↗

For the following exercises, plot the complex number in the complex plane.

$- 1 + 3 i$

Answer

Plot of -1 + 3i in the complex plane (-1 along the real axis, 3 along the imaginary).

10.R-45 §10.R source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

${\begin{cases} x (t) = - \cos t \\ y (t) = 2 \sin^{2} t \end{cases}$

Answer

x^{2} + \frac{1}{2} y = 1

10.R-47 §10.R source ↗

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

Parameterize the line from $(- 2, 3)$ to $(4, 7)$ so that the line is at $(- 2, 3)$ at $t = 0$ and $(4, 7)$ at $t = 1.$

Answer

{\begin{array}{l} x (t) = - 2 + 6 t \\ y (t) = 3 + 4 t \end{array}

10.R-49 §10.R source ↗

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

${\begin{array}{l} x (t) = e^{t} \\ y (t) = - 2 e^{5 t} \end{array}$

Answer

$y = - 2 x^{5}$

10.R-51 §10.R source ↗

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

ⓐ Find the parametric equations to model the path of the ball.

ⓑ Where is the ball after 3 seconds?

Answer

a. ${\begin{cases} x (t) = (80 \cos (40°)) t \\ y (t) = - 16 t^{2} + (80 \sin (40°)) t + 4 \end{cases}$

b. The ball is 14 feet high and 184 feet from where it was launched.

c. 3.3 seconds

10.R-53 §10.R source ↗

For the following exercises, determine whether the two vectors, $u$ and $v,$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2},$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4} .$

$P_{1} =$ $(6, 11),$ $P_{2} =$ $(- 2, 8),$ $P_{3} =$ $(0, - 1)$ and $P_{4} = (- 8, 2)$

Answer

not equal

10.R-55 §10.R source ↗

For the following exercises, use the vectors $u = 2 i -$ $j, v =$ $4 i -$ $3 j,$ and $w = - 2 i + 5 j$ to evaluate the expression.

2v − u + w

Answer

10.R-57 §10.R source ↗

For the following exercises, find a unit vector in the same direction as the given vector.

b = −3i − j

Answer

- \frac{3 \sqrt{10}}{10}

- \frac{\sqrt{10}}{10}

10.R-59 §10.R source ↗

For the following exercises, find the magnitude and direction of the vector.

$〈 −3, −3 〉$

Answer

Magnitude:

3 \sqrt{2},

Direction:

225°

10.R-61 §10.R source ↗

For the following exercises, calculate $u \cdot v .$

u = i + 4j and v = 4i + 3j

Answer

16

10.R-63 §10.R source ↗

For the following exercises, calculate $u \cdot v .$

Given the vectors shown in Figure 4, sketch u + v, u − v and 3v.

Diagram of vectors v, 2v, and 1/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1/2 v vector is in the same direction as v but has half the magnitude.

Figure 4

Answer

Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).

10.T-1 §10.T source ↗

Assume

α

is opposite side

a, β

is opposite side

b,

and

γ

is opposite side

c .

Solve the triangle, if possible, and round each answer to the nearest tenth, given

β =

68°,

b =

21,

c =

16.

Answer

α =

67.1°,

γ =

44.9°,

a =

20.9

10.T-3 §10.T source ↗

A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15° to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?

Answer

1712 miles

10.T-5 §10.T source ↗

Convert

(2, \frac{π}{3})

to rectangular coordinates.

Answer

(1, \sqrt{3})

10.T-7 §10.T source ↗

Convert to rectangular form and graph:

r = - 3 \csc θ .

Answer

$y = - 3$

Plot of the given equation in rectangular form - line y=-3.

10.T-9 §10.T source ↗

Graph

r = 3 + 3 \cos θ .

Answer

Graph of the given equations - a cardioid.

10.T-11 §10.T source ↗

Find the absolute value of the complex number

5 - 9 i .

Answer

\sqrt{106}

10.T-13 §10.T source ↗

Convert the complex number from polar to rectangular form:

z =

5 cis (\frac{2 π}{3}) .

Answer

\frac{- 5}{2} + i \frac{5 \sqrt{3}}{2}

10.T-15 §10.T source ↗

Given $z_{1} =$ $8 cis (36°)$ and $z_{2} =$ $2 cis (15°),$ evaluate each expression.

$\frac{z_{1}}{z_{2}}$

Answer

4 cis (21°)

10.T-17 §10.T source ↗

Given $z_{1} =$ $8 cis (36°)$ and $z_{2} =$ $2 cis (15°),$ evaluate each expression.

$\sqrt{z_{1}}$

Answer

2 \sqrt{2} cis (18°),

2 \sqrt{2} cis (198°)

10.T-19 §10.T source ↗

Given $z_{1} =$ $8 cis (36°)$ and $z_{2} =$ $2 cis (15°),$ evaluate each expression.

Eliminate the parameter $t$ to rewrite the following parametric equations as a Cartesian equation: ${\begin{array}{l} x (t) = t + 1 \\ y (t) = 2 t^{2} \end{array} .$

Answer

y = 2 {(x - 1)}^{2}

10.T-21 §10.T source ↗

Given $z_{1} =$ $8 cis (36°)$ and $z_{2} =$ $2 cis (15°),$ evaluate each expression.

Graph the set of parametric equations and find the Cartesian equation: ${\begin{array}{l} x (t) = - 2 \sin t \\ y (t) = 5 \cos t \end{array} .$

Answer

10.T-23 §10.T source ↗

For the following exercises, use the vectors u = i − 3j and v = 2i + 3j.

Find 2u − 3v.

Answer

−4i − 15j

10.T-25 §10.T source ↗

For the following exercises, use the vectors u = i − 3j and v = 2i + 3j.

Find a unit vector in the same direction as $v .$

Answer

\frac{2 \sqrt{13}}{13} i +

\frac{3 \sqrt{13}}{13} j

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Problems and answers are adapted from Algebra and Trigonometry 2e by Jay Abramson, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.