Chapter 7

Find the exact functional value without using a calculator: $$\cos ^{-1}\left(-\frac{1}{2}\right)$$

Rewrite the given expression in terms of \(\sin x\) and \(\cos x\). $$\sin \left(\frac{\pi}{2}+x\right)$$

Approximate all solutions in \([0,2 \pi)\) of the given equation. $$\tan x=4$$

Use the half-angle identities to evaluate the given expression exactly. $$\cos \frac{3 \pi}{8}$$

Find the exact functional value without using a calculator: $$\sin ^{-1}\left(-\frac{1}{2}\right)$$

Rewrite the given expression in terms of \(\sin x\) and \(\cos x\). $$\cos \left(x+\frac{\pi}{2}\right)$$

Approximate all solutions in \([0,2 \pi)\) of the given equation. $$\tan x=18$$

Use the half-angle identities to evaluate the given expression exactly. $$\tan \frac{\pi}{12}$$

Use a calculator in radian mode to approximate the functional value. $$\sin ^{-1} .35$$

Use your knowledge of special values to find the exact solutions of the equation. $$\sin x=\sqrt{3} / 2$$

Rewrite the given expression in terms of \(\sin x\) and \(\cos x\). $$\cos \left(x-\frac{3 \pi}{2}\right)$$

Prove the identity. $$\frac{\tan x}{\sec x}=\sin x$$

Use the half-angle identities to evaluate the given expression exactly. $$\sin \frac{5 \pi}{8}$$

Use a calculator in radian mode to approximate the functional value. $$\cos ^{-1} .76$$

Use your knowledge of special values to find the exact solutions of the equation. $$2 \cos x=\sqrt{2}$$

Rewrite the given expression in terms of \(\sin x\) and \(\cos x\). $$\csc \left(x+\frac{\pi}{2}\right)$$

Prove the identity. $$\frac{\cot x}{\csc x}=\cos x$$

Use the half-angle identities to evaluate the given expression exactly. $$\cos \frac{\pi}{12}$$

Use a calculator in radian mode to approximate the functional value. $$\tan ^{-1}(-3.256)$$

Use your knowledge of special values to find the exact solutions of the equation. $$\tan x=-\sqrt{3}$$

Prove the identity. $$(1+\cos x)(1-\cos x)=\sin ^{2} x$$

Use the half-angle identities to evaluate the given expression exactly. $$\tan \frac{5 \pi}{8}$$

Use a calculator in radian mode to approximate the functional value. $$\sin ^{-1}(-.795)$$

Use your knowledge of special values to find the exact solutions of the equation. $$\tan x=1$$

Prove the identity. $$(\csc x-1)(\csc x+1)=\cot ^{2} x$$

Use the half-angle identities to evaluate the given expression exactly. $$\sin \frac{7 \pi}{8}$$

Use a calculator in radian mode to approximate the functional value. $$\sin ^{-1}(\sin 7)[\text { The answer is not } 7 .]$$

Simplify the given expression. $$\sin 3 \cos 5-\cos 3 \sin 5$$

Use your knowledge of special values to find the exact solutions of the equation. $$2 \cos x=-\sqrt{3}$$

Use the half-angle identities to evaluate the given expression exactly. $$\cos \frac{7 \pi}{8}$$

Use a calculator in radian mode to approximate the functional value. $$\cos ^{-1}(\cos 3.5)$$

Simplify the given expression. $$\sin 37^{\circ} \sin 53^{\circ}-\cos 37^{\circ} \cos 53^{\circ}$$

Use your knowledge of special values to find the exact solutions of the equation. $$\sin x=0$$

Use a calculator in radian mode to approximate the functional value. $$\tan ^{-1}[\tan (-4)]$$

Simplify the given expression. $$\cos (x+y) \cos y+\sin (x+y) \sin y$$

Use your knowledge of special values to find the exact solutions of the equation. $$2 \sin x+1=0$$

Use the half-angle identities to evaluate the given expression exactly. $$\cot \frac{\pi}{8}$$

Use a calculator in radian mode to approximate the functional value. $$\sin ^{-1}[\sin (-2)]$$

Simplify the given expression. $$\sin (x-y) \cos y+\cos (x-y) \sin y$$

Use your knowledge of special values to find the exact solutions of the equation. $$\csc x=\sqrt{2}$$

Prove the identity. $$\tan x(\cos x+\csc x)=\sin x+\sec x$$

Use the half-angle identities to evaluate the given expression exactly. $$\cos \frac{\pi}{16}[\text {Hint}: \text { Exercise } 11]$$

Use a calculator in radian mode to approximate the functional value. $$\cos ^{-1}[\cos (-8.5)]$$

Simplify the given expression. $$\cos (x+y)-\cos (x-y)$$

Use your knowledge of special values to find the exact solutions of the equation. $$\csc x=2$$

Use the half-angle identities to evaluate the given expression exactly. $$\sin \frac{\pi}{16}$$

Let \(u\) be a number such that \(-\frac{\pi}{2} \leq u \leq \frac{\pi}{2}\) Prove that \(0 \leq \frac{\pi}{2}-u \leq \pi\)

Simplify the given expression. $$\sin (x+y)-\sin (x-y)$$

Use your knowledge of special values to find the exact solutions of the equation. $$-2 \sec x=4$$

Use the half-angle identities to evaluate the given expression exactly. $$\sin \frac{\pi}{24}[\text {Hint}: \text { Exercise } 17]$$