Chapter 5

Find all solutions of each equation. $$\cos x=\frac{\sqrt{3}}{2}$$

In Exercises \(7-14,\) use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\cot \theta=3, \theta\) lies in quadrant III.

Verify each identity. $$\cos \left(x-\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x)$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\cos 4 x+\cos 2 x$$

Verify each identity. $$\frac{\tan \theta \cot \theta}{\csc \theta}=\sin \theta$$

Find all solutions of each equation. $$\tan x=1$$

In Exercises \(7-14,\) use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\sin \theta=-\frac{9}{41}, \theta\) lies in quadrant III.

Find the exact value of each expression. $$\sin \left(45^{\circ}-30^{\circ}\right)$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\cos 9 x-\cos 7 x$$

Verify each identity. $$\frac{\cos \theta \sec \theta}{\cot \theta}=\tan \theta$$

Find all solutions of each equation. $$\tan x=\sqrt{3}$$

In Exercises \(7-14,\) use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\sin \theta=-\frac{2}{3}, \theta\) lies in quadrant III.

Find the exact value of each expression. $$\sin \left(60^{\circ}-45^{\circ}\right)$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\sin x+\sin 2 x$$

Verify each identity. $$\sin ^{2} \theta\left(1+\cot ^{2} \theta\right)=1$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$2 \sin 15^{\circ} \cos 15^{\circ}$$

Find all solutions of each equation. $$\cos x=-\frac{1}{2}$$

Find the exact value of each expression. $$\sin 105^{\circ}$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\sin x-\sin 2 x$$

Verify each identity. $$\cos ^{2} \theta\left(1+\tan ^{2} \theta\right)=1$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$2 \sin 22.5^{\circ} \cos 22.5^{\circ}$$

Find all solutions of each equation. $$\sin x=-\frac{\sqrt{2}}{2}$$

Find the exact value of each expression. $$\sin 75^{\circ}$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\cos \frac{3 x}{2}+\cos \frac{x}{2}$$

Verify each identity. $$\sin t \tan t=\frac{1-\cos ^{2} t}{\cos t}$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$\cos ^{2} 75^{\circ}-\sin ^{2} 75^{\circ}$$

Find all solutions of each equation. $$\tan x=0$$

Find the exact value of each expression. $$\cos \left(135^{\circ}+30^{\circ}\right)$$

Verify each identity. $$\cos t \cot t=\frac{1-\sin ^{2} t}{\sin t}$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$\cos ^{2} 105^{\circ}-\sin ^{2} 105^{\circ}$$

Find all solutions of each equation. $$\sin x=0$$

Find the exact value of each expression. $$\cos \left(240^{\circ}+45^{\circ}\right)$$

Verify each identity. $$\frac{\csc ^{2} t}{\cot t}=\csc t \sec t$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$2 \cos ^{2} \frac{\pi}{8}-1$$

Find all solutions of each equation. $$2 \cos x+\sqrt{3}=0$$

Find the exact value of each expression. $$\cos 75^{\circ}$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\cos 75^{\circ}-\cos 15^{\circ}$$

Verify each identity. $$\frac{\sec ^{2} t}{\tan t}=\sec t \csc t$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$1-2 \sin ^{2} \frac{\pi}{12}$$

Find all solutions of each equation. $$2 \sin x+\sqrt{3}=0$$

Find the exact value of each expression. $$\cos 105^{\circ}$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\sin \frac{\pi}{12}-\sin \frac{5 \pi}{12}$$

Verify each identity. $$\frac{\tan ^{2} t}{\sec t}=\sec t-\cos t$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$\frac{2 \tan \frac{\pi}{12}}{1-\tan ^{2} \frac{\pi}{12}}$$

Find all solutions of each equation. $$4 \sin \theta-1=2 \sin \theta$$

Find the exact value of each expression. $$\tan \left(\frac{\pi}{6}+\frac{\pi}{4}\right)$$

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\cos \frac{\pi}{12}-\cos \frac{5 \pi}{12}$$

Verify each identity. $$\frac{\cot ^{2} t}{\csc t}=\csc t-\sin t$$

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$\frac{2 \tan \frac{\pi}{8}}{1-\tan ^{2} \frac{\pi}{8}}$$

Find all solutions of each equation. $$5 \sin \theta+1=3 \sin \theta$$