Content check — Algebra and Trig, Chapter 7: The Unit Circle: Sine and Cosine Functions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 7: The Unit Circle: Sine and Cosine Functions

226 problems · 49 figures

Try It #1 §7.1 source ↗

Show an angle of

240°

on a circle in standard position.

Answer

Graph of a 240-degree angle with a counterclockwise rotation.

Try It #2 §7.1 source ↗

Find the radian measure of three-fourths of a full rotation.

Answer

\frac{3 π}{2}

Try It #3 §7.1 source ↗

Convert

- \frac{3 π}{4}

radians to degrees.

Answer

−135 °

Try It #4 §7.1 source ↗

Convert

126°

to radians.

Answer

\frac{7 π}{10}

Try It #5 §7.1 source ↗

Find an angle

α

that is coterminal with an angle measuring

870°,

where

0° \leq α <

360° .

Answer

α = 150°

Try It #6 §7.1 source ↗

Find an angle

β

that is coterminal with an angle measuring

−300°

such that

0° \leq β <

360° .

Answer

β = 60°

Try It #7 §7.1 source ↗

Find an angle of measure

θ

that is coterminal with an angle of measure

- \frac{17 π}{6}

where

0 \leq θ < 2 π .

Answer

\frac{7 π}{6}

Try It #8 §7.1 source ↗

Find the arc length along a circle of radius 10 units subtended by an angle of

215° .

Answer

\frac{215 π}{18} =

37.525 units

Try It #9 §7.1 source ↗

In central pivot irrigation, which creates the field shapes similar to the image at the beginning of Equations and Inequalities, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.

Answer

1.88

Try It #10 §7.1 source ↗

A vintage vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.

Answer

\frac{3 π}{2}

rad/s

Try It #11 §7.1 source ↗

A satellite is rotating around Earth at 0.25 radian per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.

Answer

1655 kilometers per hour

7.1-1 §7.1 source ↗

Draw an angle in standard position. Label the vertex, initial side, and terminal side.

Answer

Graph of a circle with an angle inscribed, showing the initial side, terminal side, and vertex.

7.1-3 §7.1 source ↗

State what a positive or negative angle signifies, and explain how to draw each.

Answer

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

7.1-5 §7.1 source ↗

Explain the differences between linear speed and angular speed when describing motion along a circular path.

Answer

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

7.1-7 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure.

$300°$

Answer

Graph of a circle with an angle inscribed.

7.1-9 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure.

$135°$

Answer

Graph of a circle with a 135 degree angle inscribed.

7.1-11 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure.

$\frac{2 π}{3}$

Answer

Graph of a circle with a 2pi/3 radians angle inscribed.

7.1-13 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure.

$\frac{5 π}{6}$

Answer

Graph of a circle with 5pi/6 radians angle inscribed.

7.1-15 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure.

$- \frac{π}{10}$

Answer

Graph of a circle with a –pi/10 radians angle inscribed.

7.1-17 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure as well as a coterminal angle.

$−120°$

Answer

$240°$

Graph of a circle showing the equivalence of two angles.

7.1-19 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure as well as a coterminal angle.

$\frac{22 π}{3}$

Answer

$\frac{4 π}{3}$

7.1-21 §7.1 source ↗

For the following exercises, draw an angle in standard position with the given measure as well as a coterminal angle.

$- \frac{4 π}{3}$

Answer

$\frac{2 π}{3}$

7.1-23 §7.1 source ↗

For the following exercises, refer to Figure 25. Round to two decimal places.

Graph of a circle with radius of 3 inches and an angle of 140 degrees.

Figure 25

Find the area of the sector.

Answer

\frac{7 π}{2} \approx

11.00 {in}^{2}

7.1-25 §7.1 source ↗

For the following exercises, refer to Figure 26. Round to two decimal places.

Graph of a circle with angle of 2pi/5 and a radius of 4.5 cm.

Figure 26

Find the area of the sector.

Answer

\frac{81 π}{20} \approx

12.72 {cm}^{2}

7.1-27 §7.1 source ↗

For the following exercises, convert angles in radians to degrees.

$\frac{π}{9}$ radians

Answer

20°

7.1-29 §7.1 source ↗

For the following exercises, convert angles in radians to degrees.

$\frac{π}{3}$ radians

Answer

60°

7.1-31 §7.1 source ↗

For the following exercises, convert angles in radians to degrees.

$- \frac{5 π}{12}$ radians

Answer

−75°

7.1-33 §7.1 source ↗

For the following exercises, convert angles in degrees to radians.

$90°$

Answer

\frac{π}{2}

radians

7.1-35 §7.1 source ↗

For the following exercises, convert angles in degrees to radians.

$−540°$

Answer

−3 π

radians

7.1-37 §7.1 source ↗

For the following exercises, convert angles in degrees to radians.

$180°$

Answer

π

radians

7.1-39 §7.1 source ↗

For the following exercises, convert angles in degrees to radians.

$150°$

Answer

\frac{5 π}{6}

radians

7.1-41 §7.1 source ↗

For the following exercises, use the given information to find the length of a circular arc. Round to two decimal places.

Find the length of the arc of a circle of radius 5.02 miles subtended by the central angle of $\frac{π}{3} .$

Answer

\frac{5.02 π}{3} \approx 5.26

miles

7.1-43 §7.1 source ↗

For the following exercises, use the given information to find the length of a circular arc. Round to two decimal places.

Find the length of the arc of a circle of radius 10 centimeters subtended by the central angle of $50°.$

Answer

\frac{25 π}{9} \approx 8.73

centimeters

7.1-45 §7.1 source ↗

For the following exercises, use the given information to find the length of a circular arc. Round to two decimal places.

Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is $63°.$

Answer

\frac{21 π}{10} \approx 6.60

meters

7.1-47 §7.1 source ↗

For the following exercises, use the given information to find the area of the sector. Round to four decimal places.

A sector of a circle has a central angle of $30°$ and a radius of 20 cm.

Answer

104.7198 cm²

7.1-49 §7.1 source ↗

For the following exercises, use the given information to find the area of the sector. Round to four decimal places.

A sector of a circle with radius of 0.7 inches and an angle of $π$ radians.

Answer

0.7697 in²

7.1-51 §7.1 source ↗

For the following exercises, find the angle between $0°$ and $360°$ that is coterminal to the given angle.

$−110°$

Answer

250°

7.1-53 §7.1 source ↗

For the following exercises, find the angle between $0°$ and $360°$ that is coterminal to the given angle.

$1400°$

Answer

320°

7.1-55 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

$\frac{10 π}{3}$

Answer

\frac{4 π}{3}

7.1-57 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

$\frac{44 π}{9}$

Answer

\frac{8 π}{9}

7.1-59 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

Answer

1320

rad/min

210.085

RPM

7.1-61 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

A wheel of radius $14$ inches is rotating $0.5$ rad/s. What is the linear speed $v,$ the angular speed in RPM, and the angular speed in deg/s?

Answer

7

in./s, 4.77 RPM ,

28.65

deg/s

7.1-63 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per minute). Find the linear speed if the CD has diameter of 120 millimeters.

Answer

1,

809,

557.37 mm/min =

30.16 m/s

7.1-65 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

Find the distance along an arc on the surface of Earth that subtends a central angle of 5 minutes
$(1 minute = \frac{1}{60} degree)$ . The radius of Earth is 3960 miles.

Answer

5.76

miles

7.1-67 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in $20$ minutes?

Answer

120°

7.1-69 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

A city is located at 40 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

Answer

794 miles per hour

7.1-71 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

Find the linear speed of the moon if the average distance between the earth and moon is 239,000 miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour.

Answer

2,234 miles per hour

7.1-73 §7.1 source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal to the given angle.

A car travels 3 miles. Its tires make 2640 revolutions. What is the radius of a tire in inches?

Answer

11.5 inches

Try It #1 §7.2 source ↗

Given the triangle shown in Figure 4, find the value of $sin t .$

A right triangle with sides of 7, 24, and 25. Also labeled is angle t which is opposite the side labeled 7.

Figure 4

Answer

\frac{7}{25}

Try It #2 §7.2 source ↗

Using the triangle shown in Figure 7,evaluate $sin t,$ $cos t,$ $tan t,$ $sec t,$ $csc t,$ $and cot t .$

Right triangle with sides 33, 56, and 65. Angle t is also labeled which is opposite to the side labeled 33.

Figure 7

Answer

\begin{matrix} sin t & = & \frac{33}{65}, & cos t & = & \frac{56}{65}, & tan t & = & \frac{33}{56}, \\ sec t & = & \frac{65}{56}, & csc t & = & \frac{65}{33}, & cot t & = & \frac{56}{33} \end{matrix}

Try It #3 §7.2 source ↗

Find the exact value of the trigonometric functions of

\frac{π}{4},

using side lengths.

Answer

\begin{matrix} sin (\frac{π}{4}) & = & \frac{\sqrt{2}}{2}, cos (\frac{π}{4}) = \frac{\sqrt{2}}{2}, \tan (\frac{π}{4}) = 1, \\ sec (\frac{π}{4}) & = & \sqrt{2}, csc (\frac{π}{4}) = \sqrt{2}, cot (\frac{π}{4}) = 1 \end{matrix}

Try It #4 §7.2 source ↗

\csc (\frac{π}{6}) = 2,

find

sec (\frac{π}{3}) .

Answer

Try It #5 §7.2 source ↗

A right triangle has one angle of

\frac{π}{3}

and a hypotenuse of 20. Find the unknown sides and angle of the triangle.

Answer

adjacent =

10;

opposite =

10 \sqrt{3};

missing angle is

\frac{π}{6}

Try It #6 §7.2 source ↗

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of

\frac{5 π}{12}

with the ground? Round to the nearest foot.

Answer

About 52 ft

7.2-1 §7.2 source ↗

For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

Answer

A right triangle with side opposite, adjacent, and hypotenuse labeled.

7.2-3 §7.2 source ↗

The tangent of an angle compares which sides of the right triangle?

Answer

The tangent of an angle is the ratio of the opposite side to the adjacent side.

7.2-5 §7.2 source ↗

Explain the cofunction identity.

Answer

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

7.2-7 §7.2 source ↗

For the following exercises, use cofunctions of complementary angles.

$\cos (\frac{π}{3}) =$ $\sin (___)$

Answer

\frac{π}{6}

7.2-9 §7.2 source ↗

For the following exercises, use cofunctions of complementary angles.

$\tan (\frac{π}{4}) =$ $\cot (___)$

Answer

\frac{π}{4}

7.2-11 §7.2 source ↗

For the following exercises, find the lengths of the missing sides if side $a$ is opposite angle $A,$ side $b$ is opposite angle $B,$ and side $c$ is the hypotenuse.

$\sin B =$ $\frac{1}{2},$ $a =$ $20$

Answer

b =

\frac{20 \sqrt{3}}{3},

c =

\frac{40 \sqrt{3}}{3}

7.2-13 §7.2 source ↗

For the following exercises, find the lengths of the missing sides if side $a$ is opposite angle $A,$ side $b$ is opposite angle $B,$ and side $c$ is the hypotenuse.

$\tan A =$ $100,$ $b =$ $100$

Answer

a =

10,000,

c =

10,000.5

7.2-15 §7.2 source ↗

For the following exercises, find the lengths of the missing sides if side $a$ is opposite angle $A,$ side $b$ is opposite angle $B,$ and side $c$ is the hypotenuse.

$a = 5, ∡ A = 60°$

Answer

b =

\frac{5 \sqrt{3}}{3},

c =

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

7.2-17 §7.2 source ↗

For the following exercises, use Figure 14 to evaluate each trigonometric function of angle $A .$

A right triangle with sides 4 and 10 and angle of A labeled which is opposite the side labeled 10.

Figure 14

$\sin A$

Answer

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

7.2-19 §7.2 source ↗

For the following exercises, use Figure 14 to evaluate each trigonometric function of angle $A .$

Figure 14

$\tan A$

Answer

\frac{5}{2}

7.2-21 §7.2 source ↗

For the following exercises, use Figure 14 to evaluate each trigonometric function of angle $A .$

Figure 14

$sec A$

Answer

\frac{\sqrt{29}}{2}

7.2-23 §7.2 source ↗

For the following exercises, use Figure 15 to evaluate each trigonometric function of angle $A .$

A right triangle with sides of 10 and 8 and angle of A labeled which is opposite the side labeled 10.

Figure 15

$\sin A$

Answer

\frac{5 \sqrt{41}}{41}

7.2-25 §7.2 source ↗

For the following exercises, use Figure 15 to evaluate each trigonometric function of angle $A .$

Figure 15

$\tan A$

Answer

\frac{5}{4}

7.2-27 §7.2 source ↗

For the following exercises, use Figure 15 to evaluate each trigonometric function of angle $A .$

Figure 15

$sec A$

Answer

\frac{\sqrt{41}}{4}

7.2-29 §7.2 source ↗

For the following exercises, solve for the unknown sides of the given triangle.

A right triangle with sides of 7, b, and c labeled. Angles of B and 30 degrees also labeled. The 30 degree angle is opposite the side labeled 7.

Answer

c = 14, b = 7 \sqrt{3}

7.2-31 §7.2 source ↗

For the following exercises, solve for the unknown sides of the given triangle.

A right triangle with corners labeled A, B, and C. Hypotenuse has length of 15 times square root of 2. Angle B is 45 degrees.

Answer

a = 15, b = 15

7.2-33 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

A right triangle with sides of 7, b, and c. Angles of 35 degrees and B are also labeled.

Answer

b =

9.9970,

c =

12.2041

7.2-35 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

A right triangle with sides a, b, and 12. Angles of 10 degrees and B are also labeled.

Answer

a =

2.0838,

b =

11.8177

7.2-37 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

$b = 15, ∡ B = 15°$

Answer

a =

55.9808,

c =

57.9555

7.2-39 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

$c = 50, ∡ B = 21°$

Answer

a =

46.6790,

b =

17.9184

7.2-41 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

$b =$ $3.5,$ $∡ A =$ $78°$

Answer

a =

16.4662,

c =

16.8341

7.2-43 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

Find $x .$

A triangle with angles of 36 degrees and 50 degrees and side x. Bisector in triangle with length of 85.

Answer

188.3159

7.2-45 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

Find $x .$

A right triangle with side of 119 and angle of 26 degrees. Within right triangle there is another right triangle with angle of 70 degrees instead of 26 degrees. Difference in side length between two triangles is x.

Answer

200.6737

7.2-47 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is $43°,$ and that the angle of depression to the bottom of the tower is $31° .$ How tall is the tower?

Answer

498.3471 ft

7.2-49 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is $18°,$ and that the angle of depression to the bottom of the monument is $3° .$ How far is the person from the monument?

Answer

1060.09 ft

7.2-51 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be $36° .$ From the same location, the angle of elevation to the top of the lightning rod is measured to be $38° .$ Find the height of the lightning rod.

Answer

27.372 ft

7.2-53 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

A 23-ft ladder leans against a building so that the angle between the ground and the ladder is $80° .$ How high does the ladder reach up the side of the building?

Answer

22.6506 ft

7.2-55 §7.2 source ↗

For the following exercises, use a calculator to find the length of each side to four decimal places.

The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

Answer

368.7633 ft

Try It #1 §7.3 source ↗

A certain angle $t$ corresponds to a point on the unit circle at $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ as shown in Figure 5. Find $\cos t$ and $\sin t .$

Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (negative square root of 2 over 2, square root of 2 over 2).

Figure 5

Answer

\cos (t) =

-

\frac{\sqrt{2}}{2},

\sin (t) =

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

Try It #2 §7.3 source ↗

Find cosine and sine of the angle

π .

Answer

\cos (π) =

-

1,

\sin (π) =

0

Try It #3 §7.3 source ↗

\cos (t) = \frac{24}{25}

and

t

is in the fourth quadrant, find

sin (t) .

Answer

\sin (t) = - \frac{7}{25}

Try It #4 §7.3 source ↗

Evaluate

\sin (\frac{π}{3}) .

Answer

approximately 0.866025403

Try It #5 §7.3 source ↗

Find the reference angle of

\frac{5 π}{3} .

Answer

\frac{π}{3}

Try It #6 §7.3 source ↗

ⓐ Use the reference angle of $315°$ to find $\cos (315°)$ and $\sin (315°) .$

ⓑ Use the reference angle of $- \frac{π}{6}$ to find $\cos (- \frac{π}{6})$ and $\sin (- \frac{π}{6}) .$

Answer

ⓐ $cos (315°) =$ $\frac{\sqrt{2}}{2},$ $sin (315°) =$ $\frac{- \sqrt{2}}{2}$

ⓑ $cos (- \frac{π}{6}) =$ $\frac{\sqrt{3}}{2},$ $\sin (- \frac{π}{6}) =$ $-$ $\frac{1}{2}$

Try It #7 §7.3 source ↗

Find the coordinates of the point on the unit circle at an angle of

\frac{5 π}{3} .

Answer

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

7.3-1 §7.3 source ↗

Describe the unit circle.

Answer

The unit circle is a circle of radius 1 centered at the origin.

7.3-3 §7.3 source ↗

Discuss the difference between a coterminal angle and a reference angle.

Answer

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle,

t,

formed by the terminal side of the angle

t

and the horizontal axis.

7.3-5 §7.3 source ↗

Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

Answer

The sine values are equal.

7.3-7 §7.3 source ↗

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by $t$ lies.

$sin (t) > 0$ and $\cos (t) > 0$

Answer

7.3-9 §7.3 source ↗

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by $t$ lies.

$sin (t) < 0$ and $\cos (t) > 0$

Answer

7.3-11 §7.3 source ↗

For the following exercises, find the exact value of each trigonometric function.

$\sin \frac{π}{3}$

Answer

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

7.3-13 §7.3 source ↗

For the following exercises, find the exact value of each trigonometric function.

$\cos \frac{π}{3}$

Answer

\frac{1}{2}

7.3-15 §7.3 source ↗

For the following exercises, find the exact value of each trigonometric function.

$\cos \frac{π}{4}$

Answer

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

7.3-17 §7.3 source ↗

For the following exercises, find the exact value of each trigonometric function.

$\sin π$

Answer

7.3-19 §7.3 source ↗

For the following exercises, find the exact value of each trigonometric function.

$\cos π$

Answer

-1

7.3-21 §7.3 source ↗

For the following exercises, find the exact value of each trigonometric function.

$\cos \frac{π}{6}$

Answer

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

7.3-23 §7.3 source ↗

For the following exercises, state the reference angle for the given angle.

$240°$

Answer

60°

7.3-25 §7.3 source ↗

For the following exercises, state the reference angle for the given angle.

$100°$

Answer

80 °

7.3-27 §7.3 source ↗

For the following exercises, state the reference angle for the given angle.

$135°$

Answer

45°

7.3-29 §7.3 source ↗

For the following exercises, state the reference angle for the given angle.

$\frac{2 π}{3}$

Answer

\frac{π}{3}

7.3-31 §7.3 source ↗

For the following exercises, state the reference angle for the given angle.

$\frac{- 11 π}{3}$

Answer

\frac{π}{3}

7.3-33 §7.3 source ↗

For the following exercises, state the reference angle for the given angle.

$\frac{- π}{8}$

Answer

\frac{π}{8}

7.3-35 §7.3 source ↗

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.

$300°$

Answer

60°,

Quadrant IV,

sin (300°) =

-

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

\cos (300°) =

\frac{1}{2}

7.3-37 §7.3 source ↗

$135°$

Answer

45°,

Quadrant II,

sin (135°) =

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

\cos (135°) =

-

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

7.3-39 §7.3 source ↗

$120°$

Answer

60°,

Quadrant II,

sin (120°) =

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

\cos (120°) =

-

\frac{1}{2}

7.3-41 §7.3 source ↗

$150°$

Answer

30°,

Quadrant II,

sin (150°) =

\frac{1}{2}

\cos (150°) =

-

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

7.3-43 §7.3 source ↗

$\frac{7 π}{6}$

Answer

\frac{π}{6},

Quadrant III,

sin (\frac{7 π}{6}) =

-

\frac{1}{2}

\cos (\frac{7 π}{6}) =

-

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

7.3-45 §7.3 source ↗

$\frac{3 π}{4}$

Answer

\frac{π}{4},

Quadrant II,

sin (\frac{3 π}{4}) = \frac{\sqrt{2}}{2}

\cos (\frac{3 π}{4}) =

-

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

7.3-47 §7.3 source ↗

$\frac{2 π}{3}$

Answer

\frac{π}{3},

Quadrant II,

sin (\frac{2 π}{3}) = \frac{\sqrt{3}}{2}

\cos (\frac{2 π}{3}) =

-

\frac{1}{2}

7.3-49 §7.3 source ↗

$\frac{7 π}{4}$

Answer

\frac{π}{4},

Quadrant IV,

sin (\frac{7 π}{4}) =

-

\frac{\sqrt{2}}{2},

cos (\frac{7 π}{4}) =

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

7.3-51 §7.3 source ↗

For the following exercises, find the requested value.

If $cos (t) = \frac{2}{9}$ and $t$ is in the first quadrant, find $sin (t) .$

Answer

\frac{\sqrt{77}}{9}

7.3-53 §7.3 source ↗

For the following exercises, find the requested value.

If $sin (t) = - \frac{1}{4}$ and $t$ is in the third quadrant, find $cos (t) .$

Answer

- \frac{\sqrt{15}}{4}

7.3-55 §7.3 source ↗

For the following exercises, find the requested value.

Find the coordinates of the point on a circle with radius 20 corresponding to an angle of $120° .$

Answer

(−10, 10 \sqrt{3})

7.3-57 §7.3 source ↗

For the following exercises, find the requested value.

Find the coordinates of the point on a circle with radius 16 corresponding to an angle of $\frac{5 π}{9} .$

Answer

(–2.778, 15.757)

7.3-59 §7.3 source ↗

For the following exercises, find the requested value.

State the range of the sine and cosine functions.

Answer

[–1, 1]

7.3-61 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

\frac{1}{2},

\cos t =

-

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

7.3-63 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (negative square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

-

\frac{\sqrt{2}}{2},

\cos t =

-

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

7.3-65 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (-1/2, square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

\frac{\sqrt{3}}{2},

\cos t =

-

\frac{1}{2}

7.3-67 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

-

\frac{\sqrt{2}}{2},

\cos t =

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

7.3-69 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (-1,0) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

0,

\cos t =

-

1

7.3-71 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (0.803,-0.596 is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

-

0.596,

\cos t =

0.803

7.3-73 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

\frac{1}{2},

\cos t =

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

7.3-75 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Answer

\sin t =

-

\frac{1}{2},

\cos t =

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

7.3-77 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (-0.649, 0.761) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

0.761,

\cos t =

-

0.649

7.3-79 §7.3 source ↗

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $t .$

Graph of circle with angle of t inscribed. Point of (0, 1) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t =

1,

\cos t =

0

7.3-81 §7.3 source ↗

For the following exercises, use a graphing calculator to evaluate.

$\cos \frac{5 π}{9}$

Answer

−0.1736

7.3-83 §7.3 source ↗

For the following exercises, use a graphing calculator to evaluate.

$\cos \frac{π}{10}$

Answer

0.9511

7.3-85 §7.3 source ↗

For the following exercises, use a graphing calculator to evaluate.

$\cos \frac{3 π}{4}$

Answer

−0.7071

7.3-87 §7.3 source ↗

For the following exercises, use a graphing calculator to evaluate.

$\cos 98°$

Answer

−0.1392

7.3-89 §7.3 source ↗

For the following exercises, use a graphing calculator to evaluate.

$\sin 310°$

Answer

−0.7660

7.3-91 §7.3 source ↗

For the following exercises, evaluate.

$\sin (\frac{3 π}{4}) \cos (\frac{5 π}{3})$

Answer

\frac{\sqrt{2}}{4}

7.3-93 §7.3 source ↗

For the following exercises, evaluate.

$\sin (\frac{- 9 π}{4}) \cos (\frac{- π}{6})$

Answer

- \frac{\sqrt{6}}{4}

7.3-95 §7.3 source ↗

For the following exercises, evaluate.

$\sin (\frac{7 π}{4}) \cos (\frac{- 2 π}{3})$

Answer

\frac{\sqrt{2}}{4}

7.3-97 §7.3 source ↗

For the following exercises, evaluate.

$\cos (\frac{- π}{3}) \cos (\frac{π}{4})$

Answer

\frac{\sqrt{2}}{4}

7.3-99 §7.3 source ↗

For the following exercises, evaluate.

$\sin (π) \sin (\frac{π}{6})$

Answer

7.3-101 §7.3 source ↗

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point $(0, 1),$ that is, on the due north position. Assume the carousel revolves counter clockwise.

What are the coordinates of the child after 90 seconds?

Answer

(0, –1)

7.3-103 §7.3 source ↗

When will the child have coordinates $(0.707, –0.707)$ if the ride lasts 6 minutes? (There are multiple answers.)

Answer

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

Try It #1 §7.4 source ↗

The point $(\frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2})$ is on the unit circle, as shown in Figure 3. Find $\sin t,$ $\cos t,$ $\tan t,$ $\sec t,$ $\csc t,$ and $\cot t .$

This is an image of a graph of circle with angle of t inscribed with radius 1. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

Figure 3

Answer

\sin t =

-

\frac{\sqrt{2}}{2} \cos t =

\frac{\sqrt{2}}{2},

\tan t =

-

1,

s e c t =

\sqrt{2},

\csc t =

-

\sqrt{2},

\cot t =

-

1

Try It #2 §7.4 source ↗

Find

\sin t,

\cos t,

\tan t,

\sec t,

\csc t,

and

\cot t .

when

t = \frac{π}{3} .

Answer

\sin \frac{π}{3} =

\frac{\sqrt{3}}{2},

\cos \frac{π}{3} =

\frac{1}{2},

\tan \frac{π}{3} =

\sqrt{3},

s e c \frac{π}{3} =

2,

c s c \frac{π}{3} =

\frac{2 \sqrt{3}}{3},

c o t \frac{π}{3} =

\frac{\sqrt{3}}{3}

Try It #3 §7.4 source ↗

Use reference angles to find all six trigonometric functions of

- \frac{7 π}{4} .

Answer

\begin{array}{l} \sin (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \cos (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \tan (\frac{- 7 π}{4}) = 1, \\ \sec (\frac{- 7 π}{4}) = \sqrt{2}, \csc (\frac{- 7 π}{4}) = \sqrt{2}, \cot (\frac{- 7 π}{4}) = 1 \end{array}

Try It #4 §7.4 source ↗

If the cotangent of angle

t

\sqrt{3},

what is the cotangent of

- t ?

Answer

- \sqrt{3}

Try It #5 §7.4 source ↗

Evaluate

csc (\frac{7 π}{6}) .

Answer

- 2

Try It #6 §7.4 source ↗

Simplify

(\tan t) (\cos t) .

Answer

\sin t

Try It #7 §7.4 source ↗

\sec (t) = - \frac{17}{8}

and

0 <

t <

π,

find the values of the other five functions.

Answer

\begin{array}{l} \cos t = - \frac{8}{17}, \sin t = \frac{15}{17}, \tan t = - \frac{15}{8} \\ \csc t = \frac{17}{15}, \cot t = - \frac{8}{15} \end{array}

Try It #8 §7.4 source ↗

Find the values of the six trigonometric functions of angle $t$ based on Figure 10.

This is an image of a graph of circle with angle of t inscribed. Point of (0, -1) is at intersection of terminal side of angle and edge of circle.

Figure 10

Answer

\begin{array}{l} \sin t = - 1, \cos t = 0, \tan t = Undefined \\ \sec t = Undefined, \csc t = - 1, \cot t = 0 \end{array}

Try It #9 §7.4 source ↗

\sin (t) =

\frac{\sqrt{2}}{2} and \cos (t) =

\frac{\sqrt{2}}{2},

find sec (t),

csc (t),

tan (t),

and cot (t)

Answer

\sec t =

\sqrt{2},

\csc t =

\sqrt{2},

\tan t =

1,

\cot t =

1

Try It #10 §7.4 source ↗

Evaluate the cotangent of

- \frac{π}{8} .

Answer

\approx - 2.414

7.4-1 §7.4 source ↗

On an interval of

[0, 2 π),

can the sine and cosine values of a radian measure ever be equal? If so, where?

Answer

Yes, when the reference angle is

\frac{π}{4}

and the terminal side of the angle is in quadrants I and III. Thus, a

x = \frac{π}{4}, \frac{5 π}{4},

the sine and cosine values are equal.

7.4-3 §7.4 source ↗

For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Answer

Substitute the sine of the angle in for

y

in the Pythagorean Theorem

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

Solve for

x

and take the negative solution.

7.4-5 §7.4 source ↗

Tangent and cotangent have a period of

π .

What does this tell us about the output of these functions?

Answer

The outputs of tangent and cotangent will repeat every

π

units.

7.4-7 §7.4 source ↗

For the following exercises, find the exact value of each expression.

$\sec \frac{π}{6}$

Answer

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

7.4-9 §7.4 source ↗

For the following exercises, find the exact value of each expression.

$\cot \frac{π}{6}$

Answer

\sqrt{3}

7.4-11 §7.4 source ↗

For the following exercises, find the exact value of each expression.

$\sec \frac{π}{4}$

Answer

\sqrt{2}

7.4-13 §7.4 source ↗

For the following exercises, find the exact value of each expression.

$\cot \frac{π}{4}$

Answer

7.4-15 §7.4 source ↗

For the following exercises, find the exact value of each expression.

$\sec \frac{π}{3}$

Answer

7.4-17 §7.4 source ↗

For the following exercises, find the exact value of each expression.

$\cot \frac{π}{3}$

Answer

\frac{\sqrt{3}}{3}

7.4-19 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\sec \frac{7 π}{6}$

Answer

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

7.4-21 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\cot \frac{13 π}{6}$

Answer

\sqrt{3}

7.4-23 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\sec \frac{3 π}{4}$

Answer

- \sqrt{2}

7.4-25 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\cot \frac{11 π}{4}$

Answer

–1

7.4-27 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\sec \frac{4 π}{3}$

Answer

-2

7.4-29 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\cot \frac{5 π}{3}$

Answer

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

7.4-31 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\sec 300°$

Answer

7.4-33 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\cot 240°$

Answer

\frac{\sqrt{3}}{3}

7.4-35 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\sec 120°$

Answer

–2

7.4-37 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

$\cot 315°$

Answer

–1

7.4-39 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

If $cos t = - \frac{1}{3},$ and $t$ is in quadrant III, find $\sin t,$ $\sec t,$ $\csc t,$ $\tan t,$ and $\cot t .$

Answer

\sin t = - \frac{2 \sqrt{2}}{3}

\sec t = - 3

\csc t = - \frac{3 \sqrt{2}}{4}

\tan t = 2 \sqrt{2}

\cot t = \frac{\sqrt{2}}{4}

7.4-41 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

If $\sin t = \frac{\sqrt{3}}{2}$ and $\cos t = \frac{1}{2},$ find $\sec t,$ $\csc t,$ $\tan t,$ and $\cot t .$

Answer

\sec t = 2,

\csc t = \frac{2 \sqrt{3}}{3},

\tan t = \sqrt{3},

\cot t = \frac{\sqrt{3}}{3}

7.4-43 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

If $sin t = \frac{\sqrt{2}}{2},$ what is the $sin (- t) ?$

Answer

- \frac{\sqrt{2}}{2}

7.4-45 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

If $sec t = 3.1,$ what is the $sec (- t) ?$

Answer

3.1

7.4-47 §7.4 source ↗

For the following exercises, use reference angles to evaluate the expression.

If $tan t = −1.4,$ what is the $tan (- t) ?$

Answer

1.4

7.4-49 §7.4 source ↗

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

This is an image of a graph of circle with angle of t inscribed. Point of (square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t = \frac{\sqrt{2}}{2}

\cos t = \frac{\sqrt{2}}{2}

\tan t = 1

\cot t = 1

\sec t = \sqrt{2}

\csc t = \sqrt{2}

7.4-51 §7.4 source ↗

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

This is an image of a graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t = - \frac{\sqrt{3}}{2}

\cos t = - \frac{1}{2}

\tan t = \sqrt{3}

\cot t = \frac{\sqrt{3}}{3}

\sec t = - 2

\csc t = - \frac{2 \sqrt{3}}{3}

7.4-53 §7.4 source ↗

For the following exercises, use a graphing calculator to evaluate to three decimal places.

$\cot \frac{4 π}{7}$

Answer

–0.228

7.4-55 §7.4 source ↗

For the following exercises, use a graphing calculator to evaluate to three decimal places.

$\tan \frac{5 π}{8}$

Answer

–2.414

7.4-57 §7.4 source ↗

For the following exercises, use a graphing calculator to evaluate to three decimal places.

$\csc \frac{π}{4}$

Answer

1.414

7.4-59 §7.4 source ↗

For the following exercises, use a graphing calculator to evaluate to three decimal places.

$\cot 33°$

Answer

1.540

7.4-61 §7.4 source ↗

For the following exercises, use a graphing calculator to evaluate to three decimal places.

$\sec 310°$

Answer

1.556

7.4-63 §7.4 source ↗

For the following exercises, use identities to evaluate the expression.

If $\tan (t) \approx 1.3,$ and $\cos (t) \approx$ $0.61,$ find $\sin (t) .$

Answer

\sin (t) \approx 0.79

7.4-65 §7.4 source ↗

For the following exercises, use identities to evaluate the expression.

If $\cot (t) \approx$ $0.58,$ and $\cos (t) \approx 0.5,$ find $\csc (t) .$

Answer

\csc t \approx 1.16

7.4-67 §7.4 source ↗

For the following exercises, use identities to evaluate the expression.

Determine whether the function $f (x) =$ $3 \sin^{2} x \cos x +$ $\sec x$ is even, odd, or neither.

Answer

even

7.4-69 §7.4 source ↗

For the following exercises, use identities to evaluate the expression.

Determine whether the function $f (x) =$ $\csc^{2} x +$ $\sec x$ is even, odd, or neither.

Answer

even

7.4-71 §7.4 source ↗

For the following exercises, use identities to simplify the expression.

$\frac{\sec t}{\csc t}$

Answer

\frac{\sin t}{\cos t} =

\tan t

7.4-73 §7.4 source ↗

For the following exercises, use identities to simplify the expression.

The amount of sunlight in a certain city can be modeled by the function $h =$ $16 \cos (\frac{1}{500} d),$ where $h$ represents the hours of sunlight, and $d$ is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th day of the year. State the period of the function.

Answer

13.77 hours, period:

1000 π

7.4-75 §7.4 source ↗

For the following exercises, use identities to simplify the expression.

The height of a piston, $h,$ in inches, can be modeled by the equation $y = 3 \sin x + 1,$ where $x$ represents the crank angle. Find the height of the piston when the crank angle is $55° .$

Answer

3.46 inches

7.R-1 §7.R source ↗

For the following exercises, convert the angle measures to degrees.

$\frac{π}{4}$

Answer

45°

7.R-3 §7.R source ↗

For the following exercises, convert the angle measures to radians.

$−210°$

Answer

- \frac{7 π}{6}

7.R-5 §7.R source ↗

For the following exercises, convert the angle measures to radians.

Find the length of an arc in a circle of radius 7 meters subtended by the central angle of $85° .$

Answer

10.385 meters

7.R-7 §7.R source ↗

For the following exercises, find the angle between $0°$ and $360°$ that is coterminal with the given angle.

$420°$

Answer

60°

7.R-9 §7.R source ↗

For the following exercises, find the angle between 0 and $2 π$ in radians that is coterminal with the given angle.

$- \frac{20 π}{11}$

Answer

\frac{2 π}{11}

7.R-11 §7.R source ↗

For the following exercises, draw the angle provided in standard position on the Cartesian plane.

$−210°$

Answer

This is an image of a graph of a circle with a negative angle inscribed.

7.R-13 §7.R source ↗

For the following exercises, draw the angle provided in standard position on the Cartesian plane.

$\frac{5 π}{4}$

Answer

This is an image of a graph of a circle with an angle inscribed.

7.R-15 §7.R source ↗

For the following exercises, draw the angle provided in standard position on the Cartesian plane.

Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour. Round to the nearest hundredth.

Answer

1036.73 miles per hour

7.R-17 §7.R source ↗

For the following exercises, use side lengths to evaluate.

$\cos \frac{π}{4}$

Answer

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

7.R-19 §7.R source ↗

For the following exercises, use side lengths to evaluate.

$\tan \frac{π}{6}$

Answer

\frac{\sqrt{3}}{3}

7.R-21 §7.R source ↗

For the following exercises, use side lengths to evaluate.

$\csc (18°) =$ $\sec (___°)$

Answer

72°

7.R-23 §7.R source ↗

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.

$\tan A = \frac{5}{9}, b = 6$

Answer

a =

\frac{10}{3},

c =

\frac{2 \sqrt{106}}{3}

7.R-25 §7.R source ↗

For the following exercises, use Figure 1 to evaluate each trigonometric function.

A right triangle with side lengths of 11 and 6. Corners A and B are also labeled. The angle A is opposite the side labeled 11. The angle B is opposite the side labeled 6.

Figure 1

$\tan B$

Answer

\frac{6}{11}

7.R-27 §7.R source ↗

For the following exercises, solve for the unknown sides of the given triangle.

Answer

a = \frac{5 \sqrt{3}}{2}, b = \frac{5}{2}

7.R-29 §7.R source ↗

For the following exercises, solve for the unknown sides of the given triangle.

The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. Find the answer to four decimal places.

Answer

369.2136 ft

7.R-31 §7.R source ↗

For the following exercises, solve for the unknown sides of the given triangle.

Find the exact value of $\cos \frac{π}{4} .$

Answer

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

7.R-33 §7.R source ↗

For the following exercises, solve for the unknown sides of the given triangle.

State the reference angle for $300° .$

Answer

60°

7.R-35 §7.R source ↗

For the following exercises, solve for the unknown sides of the given triangle.

Compute cosine of $330° .$

Answer

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

7.R-37 §7.R source ↗

For the following exercises, solve for the unknown sides of the given triangle.

State the domain of the sine and cosine functions.

Answer

all real numbers

7.R-39 §7.R source ↗

For the following exercises, find the exact value of the given expression.

$\cos \frac{π}{6}$

Answer

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

7.R-41 §7.R source ↗

For the following exercises, find the exact value of the given expression.

$\csc \frac{π}{3}$

Answer

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

7.R-43 §7.R source ↗

For the following exercises, use reference angles to evaluate the given expression.

$\sec \frac{11 π}{3}$

Answer

7.R-45 §7.R source ↗

For the following exercises, use reference angles to evaluate the given expression.

If $\sec (t) =$ $−2.5,$ what is the $sec (- t) ?$

Answer

–2.5

7.R-47 §7.R source ↗

For the following exercises, use reference angles to evaluate the given expression.

If $tan (t) = \frac{1}{3},$ find $tan (t - π) .$

Answer

\frac{1}{3}

7.R-49 §7.R source ↗

For the following exercises, use reference angles to evaluate the given expression.

Which trigonometric functions are even?

Answer

cosine, secant

7.T-1 §7.T source ↗

Convert

\frac{5 π}{6}

radians to degrees.

Answer

150°

7.T-3 §7.T source ↗

Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of

30° .

Answer

6.283 centimeters

7.T-5 §7.T source ↗

Find the angle between

0°

and

360°

that is coterminal with

375° .

Answer

15°

7.T-7 §7.T source ↗

Draw the angle

315°

in standard position on the Cartesian plane.

Answer

7.T-9 §7.T source ↗

A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?

Answer

3.351 feet per second,

\frac{2 π}{75}

radians per second

7.T-11 §7.T source ↗

Find the missing sides of the triangle.

A right triangle with hypotenuse length of 9 and angle measure of 60 degrees.

Answer

a = \frac{9}{2}, b = \frac{9 \sqrt{3}}{2}

7.T-13 §7.T source ↗

Find the exact value of

\sin \frac{π}{6} .

Answer

\frac{1}{2}

7.T-15 §7.T source ↗

State the domain of the sine and cosine functions.

Answer

real numbers

7.T-17 §7.T source ↗

Find the exact value of

\cot \frac{π}{4} .

Answer

7.T-19 §7.T source ↗

Use reference angles to evaluate

\csc \frac{7 π}{4} .

Answer

- \sqrt{2}

7.T-21 §7.T source ↗

csc t = 0.68,

what is the

csc (- t) ?

Answer

–0.68

7.T-23 §7.T source ↗

Find the missing angle:

\cos (\frac{π}{6}) =

\sin (___)

Answer

\frac{π}{3}

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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.