Chapter 5

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\sin 6 x \sin 2 x$$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos x=\frac{\sqrt{2}}{2}, \quad x=\frac{\pi}{4}$$

Verify each identity. $$\sin x \sec x=\tan x$$

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

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\sin 8 x \sin 4 x$$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\tan x=\sqrt{3}, \quad x=\frac{\pi}{3}$$

Verify each identity. $$\cos x \csc x=\cot x$$

Find the exact value of each expression.. $$\cos \left(120^{\circ}-45^{\circ}\right)$$

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\cos 7 x \cos 3 x$$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\sin x=\frac{\sqrt{3}}{2}, \quad x=\frac{\pi}{6}$$

Verify each identity. $$\tan (-x) \cos x=-\sin x$$

Find the exact value of each expression.. $$\cos \left(\frac{3 \pi}{4}-\frac{\pi}{6}\right)$$

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\cos 9 x \cos 2 x$$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\sin x=\frac{\sqrt{2}}{2}, \quad x=\frac{\pi}{3}$$

Verify each identity. $$\cot (-x) \sin x=-\cos x$$

Find the exact value of each expression.. $$\cos \left(\frac{2 \pi}{3}-\frac{\pi}{6}\right)$$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos x=-\frac{1}{2}, \quad x=\frac{2 \pi}{3}$$

Verify each identity. $$\tan x \csc x \cos x=1$$

Expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\). a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$\cos 50^{\circ} \cos 20^{\circ}+\sin 50^{\circ} \sin 20^{\circ}$$

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\sin 2 x \cos 3 x$$

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos x=-\frac{1}{2}, \quad x=\frac{4 \pi}{3}$$

Verify each identity. $$\cot x \sec x \sin x=1$$

Expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\). a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$\cos 50^{\circ} \cos 5^{\circ}+\sin 50^{\circ} \sin 5^{\circ}$$

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\cos \frac{3 x}{2} \sin \frac{x}{2}$$

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. \(\tan 2 \theta\) \(\sin \theta=\frac{15}{17}, \theta\) lies in quadrant II.

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\tan 2 x=-\frac{\sqrt{3}}{3}, \quad x=\frac{5 \pi}{12}$$

Verify each identity. $$\sec x-\sec x \sin ^{2} x=\cos x$$

Expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\). a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$\cos \frac{5 \pi}{12} \cos \frac{\pi}{12}+\sin \frac{5 \pi}{12} \sin \frac{\pi}{12}$$

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\cos \frac{5 x}{2} \sin \frac{x}{2}$$

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. \(\tan 2 \theta\) \(\sin \theta=\frac{12}{13}, \theta\) lies in quadrant II.

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos \frac{2 x}{3}=-\frac{1}{2}, \quad x=\pi$$

Verify each identity. $$\csc x-\csc x \cos ^{2} x=\sin x$$

Expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\). a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$\cos \frac{5 \pi}{18} \cos \frac{\pi}{9}+\sin \frac{5 \pi}{18} \sin \frac{\pi}{9}$$

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. $$\sin 6 x+\sin 2 x$$

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. \(\tan 2 \theta\) \(\cos \theta=\frac{24}{25}, \theta\) lies in quadrant IV.

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos x=\sin 2 x, \quad x=\frac{\pi}{3}$$

Verify each identity. $$\cos ^{2} x-\sin ^{2} x=1-2 \sin ^{2} x$$

Verify each identity. $$\frac{\cos (\alpha-\beta)}{\cos \alpha \sin \beta}=\tan \alpha+\cot \beta$$

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. $$\sin 8 x+\sin 2 x$$

Verify each identity. $$\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1$$

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. \(\tan 2 \theta\) \(\cos \theta=\frac{40}{41}, \theta\) lies in quadrant IV.

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos x+2=\sqrt{3} \sin x, \quad x=\frac{\pi}{6}$$

Verify each identity. $$\frac{\cos (\alpha-\beta)}{\sin \alpha \sin \beta}=\cot \alpha \cot \beta+1$$

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. $$\sin 7 x-\sin 3 x$$

Verify each identity. $$\csc \theta-\sin \theta=\cot \theta \cos \theta$$

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. \(\tan 2 \theta\) \(\cot \theta=2, \theta\) lies in quadrant III.

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

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

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. $$\sin 11 x-\sin 5 x$$

Verify each identity. $$\tan \theta+\cot \theta=\sec \theta \csc \theta$$