Chapter 5: Trigonometric FunctionsContent check — Precalculus, Chapter 5: Trigonometric Functions

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

300°

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

135°

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

$\frac{2\pi }{3}$

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

$\frac{5\pi }{6}$

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

$-\frac{\pi }{10}$

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

−120°

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

$\frac{22\pi }{3}$

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

$-\frac{4\pi }{3}$

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

Find the area of the sector.

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

Find the area of the sector.

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

$\frac{\pi }{9}$ radians

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

$\frac{\pi }{3}$ radians

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

$-\frac{5\pi }{12}$ radians

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

90°

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

−540°

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

180°

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

150°

For the following exercises, use to 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 $\frac{\pi }{3}.$

For the following exercises, use to 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°.

For the following exercises, use to 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°.

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.

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 $\pi$ radians.

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

−110°

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

1400°

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

$\frac{10\pi }{3}$

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

$\frac{44\pi }{9}$

For the following exercises, find the angle between 0 and $2\pi$ 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?

For the following exercises, find the angle between 0 and $2\pi$ 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?

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

When being burned in a writable CD-R drive, the angular speed of a CD 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 4,800 RPM (revolutions per minute. Find the linear speed if the CD has diameter of 120millimeters.

For the following exercises, find the angle between 0 and $2\pi$ 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 $\left(1\text{ minute}=\frac{1}{60}\text{ degree}\right)$. The radius of Earth is 3960 miles.

For the following exercises, find the angle between 0 and $2\pi$ 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?

For the following exercises, find the angle between 0 and $2\pi$ 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. Refer to Unit Circle: Sine and Cosine Functions for information on trigonometric functions.

For the following exercises, find the angle between 0 and $2\pi$ 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.

For the following exercises, find the angle between 0 and $2\pi$ 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?

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

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

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

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.

$\text{sin}(t)>0$ and $\cos (t)>0$

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 $\cos (t)>0$

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

$\sin \frac{\pi }{3}$

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

$\cos \frac{\pi }{3}$

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

$\cos \frac{\pi }{4}$

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

$\sin \pi$

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

$\cos \pi$

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

$\cos \frac{\pi }{6}$

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

$\mathrm{240°}$

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

$\mathrm{100°}$

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

$\mathrm{135°}$

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

$\frac{2\pi }{3}$

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

$\frac{-11\pi }{3}$

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

$\frac{-\pi }{8}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\mathrm{300°}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\mathrm{135°}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\mathrm{120°}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\mathrm{150°}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{7\pi }{6}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{3\pi }{4}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{2\pi }{3}$

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 special angles on the unit circle, use a calculator and round to three decimal places.

$\frac{7\pi }{4}$

For the following exercises, find the requested value.

If $\text{cos}\left(t\right)=\frac{2}{9}$ and $t$ is in the 1^st quadrant, find $\text{sin}(t).$

For the following exercises, find the requested value.

If $\text{sin}\left(t\right)=-\frac{1}{4}$ and $t$ is in the 3^rd quadrant, find $\text{cos}(t).$

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 $\mathrm{120°}.$

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 $\frac{5\pi }{9}.$

For the following exercises, find the requested value.

State the range of the sine and cosine functions.

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

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

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

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

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

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

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

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

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

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

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

$\cos \frac{5\pi }{9}$

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

$\cos \frac{\pi }{10}$

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

$\cos \frac{3\pi }{4}$

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

$\cos \mathrm{98°}$

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

$\sin \mathrm{310°}$

For the following exercises, evaluate.

$\sin \left(\frac{3\pi }{4}\right)\cos \left(\frac{5\pi }{3}\right)$

For the following exercises, evaluate.

$\sin \left(\frac{-9\pi }{4}\right)\cos \left(\frac{-\pi }{6}\right)$

For the following exercises, evaluate.

$\sin \left(\frac{7\pi }{4}\right)\cos \left(\frac{-2\pi }{3}\right)$

For the following exercises, evaluate.

$\cos \left(\frac{-\pi }{3}\right)\cos \left(\frac{\pi }{4}\right)$

For the following exercises, evaluate.

$\sin \left(\pi \right)\sin \left(\frac{\pi }{6}\right)$

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 $\left(0,1\right),$ that is, on the due north position. Assume the carousel revolves counter clockwise.

What are the coordinates of the child after 90 seconds?

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 $\left(0,1\right),$ that is, on the due north position. Assume the carousel revolves counter clockwise.

When will the child have coordinates $\left(0.707,\mathrm{–0.707}\right)$ if the ride lasts 6 minutes? (There are multiple answers.)

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

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

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

$\sec \frac{\pi }{6}$

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

$\cot \frac{\pi }{6}$

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

$\sec \frac{\pi }{4}$

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

$\cot \frac{\pi }{4}$

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

$\sec \frac{\pi }{3}$

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

$\cot \frac{\pi }{3}$

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

$\sec \frac{7\pi }{6}$

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

$\cot \frac{13\pi }{6}$

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

$\sec \frac{3\pi }{4}$

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

$\cot \frac{11\pi }{4}$

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

$\sec \frac{4\pi }{3}$

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

$\cot \frac{5\pi }{3}$

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

$\sec \mathrm{300°}$

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

$\cot \mathrm{240°}$

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

$\sec \mathrm{120°}$

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

$\cot \mathrm{315°}$

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

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

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 $\cot t.$

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

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

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

If $\text{sec}t=3.1,$ what is the $\text{sec}(-t)?$

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

If $\text{tan}t=-1.4,$ what is the $\text{tan}(-t)?$

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

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

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

$\cot \frac{4\pi }{7}$

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

$\tan \frac{5\pi }{8}$

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

$\csc \frac{\pi }{4}$

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

$\cot \mathrm{33°}$

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

$\sec \mathrm{310°}$

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

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

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

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

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.

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.

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

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

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

The amount of sunlight in a certain city can be modeled by the function $h=$ $16\cos \left(\frac{1}{500}d\right),$ 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 267^th day of the year. State the period of the function.

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

The height of a piston, $h,$ in inches, can be modeled by the equation $y=2\cos x+6,$ where $x$ represents the crank angle. Find the height of the piston when the crank angle is $30\text{°}.$

Given the triangle shown in Figure 4, find the value of $\text{sin}t.$

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

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

For the following exercises, use cofunctions of complementary angles.

$\cos \left(\frac{\pi }{3}\right)=$ $\sin \text{(\_\_\_)}$

For the following exercises, use cofunctions of complementary angles.

$\tan \left(\frac{\pi }{4}\right)=$ $\cot (\text{\_\_})$

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.

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

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.

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

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.

$a=5,\measuredangle A={60}^{\circ }$

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

$\sin A$

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

$\tan A$

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

$\sec A$

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

$\sin A$

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

$\tan A$

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

$\sec A$

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

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

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

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

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

$b=15,\measuredangle B={15}^{\circ }$

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

$c=50,\measuredangle B={21}^{\circ }$

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

$b=$ $3.5,$ $\measuredangle A=$ ${78}^{\circ }$

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

Find $x.$

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

Find $x.$

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 $\mathrm{43°},$ and that the angle of depression to the bottom of the tower is $\mathrm{31°}.$ How tall is the tower?

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 $\mathrm{18°},$ and that the angle of depression to the bottom of the monument is $\mathrm{3°}.$ How far is the person from the monument?

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 $\mathrm{36°}.$ From the same location, the angle of elevation to the top of the lightning rod is measured to be $\mathrm{38°}.$ Find the height of the lightning rod.

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 $\mathrm{80°}.$ How high does the ladder reach up the side of the building?

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.

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

$\frac{\pi }{4}$

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

-210°

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°.

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

$\mathrm{420°}$

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

$-\frac{20\pi }{11}$

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

-210°

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

$\frac{5\pi }{4}$

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.

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

Find the exact value of $\sin \frac{\pi }{3}.$

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

Find the exact value of $\cos \pi .$

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

State the reference angle for $\frac{3\pi }{4}.$

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

Compute sine of $\frac{5\pi }{4}.$

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

State the range of the sine and cosine functions.

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

$\tan \frac{\pi }{4}$

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

$\sec \frac{\pi }{4}$

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

$\sec \mathrm{315°}$

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

If $\text{tan}(t)=$ $-$ $0.6,$ what is the $\text{tan}(-t)?$