Chapter 5

Find the exact value of each expression. $$ \sin ^{-1} \frac{1}{2} $$

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=4 \sin x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (-4,3) $$

In Exercises \(1-6,\) the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. $$ 135^{\circ} $$

Find the exact value of each expression. $$ \sin ^{-1} 0 $$

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=5 \sin x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (-12,5) $$

In Exercises \(1-6,\) the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. $$ 177^{\circ} $$

Find the exact value of each expression. $$ \sin ^{-1} \frac{\sqrt{2}}{2} $$

Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. $$A=52.6^{\circ}, c=54$$

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=\frac{1}{5} \sin x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (2,3) $$

In Exercises \(1-6,\) the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. $$ 83.135^{\circ} $$

Find the exact value of each expression. $$ \sin ^{-1} \frac{\sqrt{3}}{2} $$

Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. $$A=54.8^{\circ}, c=80$$

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=\frac{1}{4} \sin x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (3,7) $$

In Exercises \(1-6,\) the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. $$ 87.177^{\circ} $$

Find the exact value of each expression. $$ \sin ^{-1}\left(-\frac{1}{2}\right) $$

In Exercises 5–12, graph two periods of the given tangent function. $$ y=3 \tan \frac{x}{4} $$

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=-3 \sin x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (3,-3) $$

In Exercises \(1-6,\) the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. $$ \pi $$

Find the exact value of each expression. $$ \sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right) $$

In Exercises 5–12, graph two periods of the given tangent function. $$ y=2 \tan \frac{x}{4} $$

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=-4 \sin x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (5,-5) $$

In Exercises \(1-6,\) the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. $$ \frac{\pi}{2} $$

Find the exact value of each expression. $$ \cos ^{-1} \frac{\sqrt{3}}{2} $$

In Exercises 5–12, graph two periods of the given tangent function. $$ y=\frac{1}{2} \tan 2 x $$

Determine the amplitude and period of each function. Then graph one period of the function. $$y=\sin 2 x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (-2,-5) $$

In Exercises \(7-12,\) find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). $$ Radius, r \quad Arc Length, s $$ $$10 inches \quad 40 inches$$

Find the exact value of each expression. $$ \cos ^{-1} \frac{\sqrt{2}}{2} $$

In Exercises 5–12, graph two periods of the given tangent function. $$ y=2 \tan 2 x $$

Determine the amplitude and period of each function. Then graph one period of the function. $$y=\sin 4 x$$

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (-1,-3) $$

In Exercises \(7-12,\) find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). $$ Radius, r \quad Arc Length, s $$ $$ 5 feet \quad 30 feet $$

Find the exact value of each expression. $$ \cos ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

In Exercises 5–12, graph two periods of the given tangent function. $$ y=-2 \tan \frac{1}{2} x $$

Determine the amplitude and period of each function. Then graph one period of the function. $$y=3 \sin \frac{1}{2} x$$

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator. $$ \cos 30^{\circ} $$

evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$ \cos \pi $$

In Exercises \(7-12,\) find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). $$ Radius, r \quad Arc Length, s $$ $$ 6 yards \quad 8 yards $$

Find the exact value of each expression. $$ \cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right) $$

In Exercises \(5-18,\) the unit circle has been divided into twelve equal arcs, corresponding to t-values of $$ 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2 \pi}{3}, \frac{5 \pi}{6}, \pi, \frac{7 \pi}{6}, \frac{4 \pi}{3}, \frac{3 \pi}{2}, \frac{5 \pi}{3}, \frac{11 \pi}{6}, \text { and } 2 \pi $$ Use the \((x, y)\) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined. $$ \tan 0 $$

In Exercises 5–12, graph two periods of the given tangent function. $$ y=-3 \tan \frac{1}{2} x $$

Determine the amplitude and period of each function. Then graph one period of the function. $$y=2 \sin \frac{1}{4} x$$

$$ \tan 30^{\circ} $$

evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$ \tan \pi $$