5.3 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 $\sin \left(\frac{5 \pi}{6}\right) = \frac{1}{2} , \text{cos} \left(\frac{5 \pi}{6}\right) = - \frac{\sqrt{3}}{2} , \text{evaluate} \sec \left(\frac{5 \pi}{6}\right) .$

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

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

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 $\sin t , \sec t , \csc t , \tan t , \cot t .$

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.

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

Use a graphing calculator to evaluate.

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

Use a graphing calculator to evaluate.

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

Use a graphing calculator to evaluate.

$\cot 33°$

Use a graphing calculator to evaluate.

$\sec 310°$

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

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