Chapter 7

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\sin x=\frac{5}{13} \quad\left(0

Find the exact functional value without using a calculator: $$\sin ^{-1} 1$$

Find all solutions of the equation. $$\sin x=.465$$

$$\text {Find the exact value.}$$ $$\cos \frac{\pi}{12}$$

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\sin x=-\frac{4}{5} \quad\left(\pi

Find the exact functional value without using a calculator: $$\cos ^{-1} 0$$

Find all solutions of the equation. $$\sin x=.682$$

$$\text {Find the exact value.}$$ $$\tan \frac{\pi}{12}$$

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\cos x=-\frac{3}{5} \quad\left(\pi

Find the exact functional value without using a calculator: $$\tan ^{-1}(-1)$$

Find all solutions of the equation. $$\cos x=-.564$$

$$\text {Find the exact value.}$$ $$\sin \frac{5 \pi}{12}$$

Test the equation graphically to determine whether it might be an identity. You need not prove those equations that seem to be identities. $$\frac{1-\cos (2 x)}{2}=\sin ^{2} x$$

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\cos x=-\frac{1}{3} \quad\left(\frac{\pi}{2}

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

Find all solutions of the equation. $$\cos x=-.371$$

$$\text {Find the exact value.}$$ $$\cos \frac{5 \pi}{12}$$

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\tan x=\frac{3}{4} \quad\left(\pi

Find the exact functional value without using a calculator: $$\cos ^{-1} 1$$

Find all solutions of the equation. $$\tan x=-.354$$

Insert one of \(A-F\) on the right of the equal sign so that the resulting equation appears to be an identity when you test it graphically. You need not prove the identity. A. \(\cos x\) B. \(\sec x\) C. \(\sin ^{2} x\) D. \(\sec ^{2} x\) E. \(\sin x-\cos x\) F. \(\frac{1}{\sin x \cos x}\) \(\csc x \tan x=\) ____________

Find the exact functional value without using a calculator: $$\tan ^{-1} 1$$

Find all solutions of the equation. $$\tan x=10$$

$$\text {Find the exact value.}$$ $$\sin \frac{7 \pi}{12}$$

Insert one of \(A-F\) on the right of the equal sign so that the resulting equation appears to be an identity when you test it graphically. You need not prove the identity. A. \(\cos x\) B. \(\sec x\) C. \(\sin ^{2} x\) D. \(\sec ^{2} x\) E. \(\sin x-\cos x\) F. \(\frac{1}{\sin x \cos x}\) \(\frac{\sin x}{\tan x}=\) _____________

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\csc x=4 \quad\left(0

Find the exact functional value without using a calculator: $$\tan ^{-1}(\sqrt{3} / 3)$$

Find all solutions of the equation. $$\cot x=2.3$$

$$\text {Find the exact value.}$$ $$\tan \frac{7 \pi}{12}$$

Insert one of \(A-F\) on the right of the equal sign so that the resulting equation appears to be an identity when you test it graphically. You need not prove the identity. A. \(\cos x\) B. \(\sec x\) C. \(\sin ^{2} x\) D. \(\sec ^{2} x\) E. \(\sin x-\cos x\) F. \(\frac{1}{\sin x \cos x}\) \(\frac{\sin ^{4} x-\cos ^{4} x}{\sin x+\cos x}=\) _____________

A batter hits a baseball that is caught by a fielder. If the ball leaves the bat at an angle of \(\theta\) radians to the horizontal, with an initial velocity of \(v\) feet per second, then the approximate horizontal distance \(d\) traveled by the ball is given by $$ d=\frac{v^{2} \sin \theta \cos \theta}{16} $$ (a) Use an identity to show that $$ d=\frac{v^{2} \sin 2 \theta}{32} $$ (b) If the initial velocity is \(115 \mathrm{ft} /\) second, what angle \(\theta\) will produce the maximum distance? [Hint: Use part (a). For what value of \(\theta \text { is } \sin 2 \theta \text { as large as possible? }]\) (figure cannot copy)

Find the exact functional value without using a calculator: $$\cos ^{-1}(\sqrt{3} / 2)$$

Find all solutions of the equation. $$\cot x=-3.5$$

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

$$\text {Find the exact value.}$$ $$\cos \frac{11 \pi}{12}$$

Prove the identity. $$\tan x \cos x=\sin x$$

Find the exact functional value without using a calculator: $$\sin ^{-1}(\sqrt{3} / 2)$$

$$\text {Find the exact value.}$$ $$\sin 75^{\circ}\left[\text {Hint}: 75^{\circ}=45^{\circ}+30^{\circ}\right]^{*}$$

Prove the identity. $$\cot x \sin x=\cos x$$

Find all solutions of the equation. $$\csc x=6.4$$

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

$$\text {Find the exact value.}$$ $$\sin 105^{\circ}$$

In Exercises \(11-14,\) approximate all solutions in \([0,2 \pi)\) of the given equation. $$\sin x=.119$$

Prove the identity. $$\cos x \sec x=1$$

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

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

$$\text {Find the exact value.}$$ $$\cos 165^{\circ}$$

Approximate all solutions in \([0,2 \pi)\) of the given equation. $$\cos x=.958$$

Prove the identity. $$\sin x \csc x=1$$

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