7.4 The Other Trigonometric Functions

Find $\sin t , \cos t , \tan t , \sec t , \csc t ,$ and $\cot t .$ when $t = \frac{\pi}{6} .$

Find $\sin t , \cos t , \tan t , \sec t , \csc t ,$ and $\cot t .$ when $t = \frac{\pi}{3} .$

Use reference angles to find all six trigonometric functions of $- \frac{5 \pi}{6} .$

Use reference angles to find all six trigonometric functions of $- \frac{7 \pi}{4} .$

If the secant of angle $t$ is 2, what is the secant of $− t ?$

If the cotangent of angle $t$ is $\sqrt{3} ,$ what is the cotangent of $− t ?$

1. ⓐ Given $\sin \left(45°\right) = \frac{\sqrt{2}}{2} , \cos \left(45°\right) = \frac{\sqrt{2}}{2} ,$ evaluate $\tan \left(45°\right) .$
2. ⓑ Given $\sin \left(\frac{5 \pi}{6}\right) = \frac{1}{2} , \cos \left(\frac{5 \pi}{6}\right) = - \frac{\sqrt{3}}{2} ,$ evaluate $\sec \left(\frac{5 \pi}{6}\right) .$

Evaluate $\text{csc} \left(\frac{7 \pi}{6}\right) .$

Simplify $\frac{\sec t}{\tan t} .$

Simplify $\left(\tan t\right) \left(\cos t\right) .$

If $\sec \left(t\right) = - \frac{17}{8}$ and $0 < t < \pi ,$ find the values of the other five functions.

If $\sin \left(t\right) = - \frac{\sqrt{3}}{2} \text{and} \text{cos} \left(t\right) = \frac{1}{2} , \text{find} \text{sec} \left(t\right) , \text{csc} \left(t\right) , \text{tan} \left(t\right) , \text{cot} \left(t\right) .$

$\sin \left(t\right) = \frac{\sqrt{2}}{2} \text{and} \cos \left(t\right) = \frac{\sqrt{2}}{2} , \text{find} \text{sec} \left(t\right) , \text{csc} \left(t\right) , \text{tan} \left(t\right) , \text{and} \text{cot} \left(t\right)$

Evaluate the cosecant of $\frac{5 \pi}{7} .$

Evaluate the cotangent of $- \frac{\pi}{8} .$

On an interval of $\left[0 , 2 \pi\right) ,$ can the sine and cosine values of a radian measure ever be equal? If so, where?

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

Tangent and cotangent have a period of $\pi .$ What does this tell us about the output of these functions?

Find the exact value of each expression.

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

Find the exact value of each expression.

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

Find the exact value of each expression.

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

Find the exact value of each expression.

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

Find the exact value of each expression.

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

Find the exact value of each expression.

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

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

$\sec 300°$

Use reference angles to evaluate the expression.

$\cot 240°$

Use reference angles to evaluate the expression.

$\sec 120°$

Use reference angles to evaluate the expression.

$\cot 315°$

Use reference angles to evaluate the expression.

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

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 $\cot t .$

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

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

Use reference angles to evaluate the expression.

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

Use a graphing calculator to evaluate to three decimal places.

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

Use a graphing calculator to evaluate to three decimal places.

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

Use a graphing calculator to evaluate to three decimal places.

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

Use a graphing calculator to evaluate to three decimal places.

$\cot 33°$

Use a graphing calculator to evaluate to three decimal places.

$\sec 310°$

Use identities to evaluate the expression.

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

Use identities to evaluate the expression.

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

Use identities to evaluate the expression.

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

Use identities to evaluate the expression.

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

Use identities to simplify the expression.

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

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 267th day of the year. State the period of the function.

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 $55° .$

Convert the angle measures to degrees.

$\frac{\pi}{4}$

Convert the angle measures to radians.

$−210°$

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 $85° .$

Find the angle between $0°$ and $\text{360}°$ that is coterminal with the given angle.

$420°$

Find the angle between 0 and $2 \pi$ in radians that is coterminal with the given angle.

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

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.

Use side lengths to evaluate.

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

Use side lengths to evaluate.

$\tan \frac{\pi}{6}$

Use side lengths to evaluate.

$\csc \left(18°\right) = \sec \left(___°\right)$

Use the given information to find the lengths of the other two sides of the right triangle.

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

$\tan B$

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.

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

State the reference angle for $300° .$

Compute cosine of $330° .$

State the domain of the sine and cosine functions.

Find the exact value of the given expression.

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

Find the exact value of the given expression.

$\csc \frac{\pi}{3}$

Use reference angles to evaluate the given expression.

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

Use reference angles to evaluate the given expression.

If $\sec \left(t\right) = −2.5 ,$ what is the $\text{sec} \left(- t\right) ?$

Use reference angles to evaluate the given expression.

If $\text{tan} \left(t\right) = \frac{1}{3} ,$ find $\text{tan} \left(t - \pi\right) .$

Use reference angles to evaluate the given expression.

Which trigonometric functions are even?

Convert $\frac{5 \pi}{6}$ radians to degrees.

Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of $30° .$

Find the angle between $0°$ and $\text{360}°$ that is coterminal with $375° .$

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?

Find the exact value of $\sin \frac{\pi}{6} .$

State the domain of the sine and cosine functions.

Find the exact value of $\cot \frac{\pi}{4} .$

Use reference angles to evaluate $\csc \frac{7 \pi}{4} .$

If $\text{csc} t = 0.68 ,$ what is the $\text{csc} \left(- t\right) ?$

Find the missing angle: $\cos \left(\frac{\pi}{6}\right) = \sin \left(___\right)$