Chapter 7

Interference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by \(f_{1}(t)=C \sin \omega t\) and \(f_{2}(t)=C \sin (\omega t+\alpha) .\) The two sound waves interfere to produce a single sound modeled by the sum of these functions $$ f(t)=C \sin \omega t+C \sin (\omega t+\alpha) $$ (a) Use the Addition Formula for Sine to show that \(f\) can be written in the form \(f(t)=A \sin \omega t+B \cos \omega t,\) where \(A\) and \(B\) are constants that depend on \(\alpha .\) (b) Suppose that \(C=10\) and \(\alpha=\pi / 3 .\) Find constants \(k\) and \(\phi\) so that \(f(t)=k \sin (\omega t+\phi)\)

Verify the identity. $$\frac{\csc x-\cot x}{\sec x-1}=\cot x$$

Find the value of the product or sum. $$\sin 75^{\circ}+\sin 15^{\circ}$$

Addition Formula for Sine In the text we proved only the Addition and Subtraction Formulas for cosine. Use these formulas and the cofunction identities $$ \begin{aligned} \sin x &=\cos \left(\frac{\pi}{2}-x\right) \\ \cos x &=\sin \left(\frac{\pi}{2}-x\right) \end{aligned} $$ to prove the Addition Formula for Sine. [Hint: To get started, use the first cofunction identity to write $$ \begin{aligned} \sin (s+t) &=\cos \left(\frac{\pi}{2}-(s+t)\right) \\ &=\cos \left(\left(\frac{\pi}{2}-s\right)-t\right) \end{aligned} $$ and use the Subtraction Formula for cosine. \(]\)

Verify the identity. $$\frac{\csc ^{2} x-\cot ^{2} x}{\sec ^{2} x}=\cos ^{2} x$$

Find the value of the product or sum. $$\cos 255^{\circ}-\cos 195^{\circ}$$

Addition Formula for Tangent Use the Addition Formulas for cosine and sine to prove the Addition Formula for Tangent. \([\)Hint: Use $$ \tan (s+t)=\frac{\sin (s+t)}{\cos (s+t)} $$ and divide the numerator and denominator by \(\cos s \cos t .]\)

Verify the identity. $$\tan ^{2} u-\sin ^{2} u=\tan ^{2} u \sin ^{2} u$$

Find the value of the product or sum. $$\cos \frac{\pi}{12}+\cos \frac{5 \pi}{12}$$

Verify the identity. $$\frac{\tan v \sin v}{\tan v+\sin v}=\frac{\tan v-\sin v}{\tan v \sin v}$$

Prove the identity. $$\cos ^{2} 5 x-\sin ^{2} 5 x=\cos 10 x$$

Verify the identity. $$\sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x$$

Prove the identity. $$\sin 8 x=2 \sin 4 x \cos 4 x$$

Verify the identity. $$\frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta$$

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

Verify the identity. $$\frac{\cos \theta}{1-\sin \theta}=\frac{\sin \theta-\csc \theta}{\cos \theta-\cot \theta}$$

Prove the identity. $$\frac{2 \tan x}{1+\tan ^{2} x}=\sin 2 x$$

Verify the identity. $$\frac{1+\tan x}{1-\tan x}=\frac{\cos x+\sin x}{\cos x-\sin x}$$

Prove the identity. $$\frac{\sin 4 x}{\sin x}=4 \cos x \cos 2 x$$

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

Verify the identity. $$\frac{1}{1-\sin x}-\frac{1}{1+\sin x}=2 \sec x \tan x$$

Prove the identity. $$\frac{2(\tan x-\cot x)}{\tan ^{2} x-\cot ^{2} x}=\sin 2 x$$

Verify the identity. $$\frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x$$

Prove the identity. $$\cot 2 x=\frac{1-\tan ^{2} x}{2 \tan x}$$

Verify the identity. $$\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x$$

Prove the identity. $$\tan 3 x=\frac{3 \tan x-\tan ^{3} x}{1-3 \tan ^{2} x}$$

Verify the identity. $$(\tan x+\cot x)^{2}=\sec ^{2} x+\csc ^{2} x$$

Prove the identity. $$4\left(\sin ^{6} x+\cos ^{6} x\right)=4-3 \sin ^{2} 2 x$$

Verify the identity. $$\tan ^{2} x-\cot ^{2} x=\sec ^{2} x-\csc ^{2} x$$

Prove the identity. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$

Verify the identity. $$\frac{\sec u-1}{\sec u+1}=\frac{1-\cos u}{1+\cos u}$$

Prove the identity. $$\tan ^{2}\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x}$$

Verify the identity. $$\frac{\cot x+1}{\cot x-1}=\frac{1+\tan x}{1-\tan x}$$

Prove the identity. $$\frac{\sin x+\sin 5 x}{\cos x+\cos 5 x}=\tan 3 x$$

Verify the identity. $$\frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x}=1-\sin x \cos x$$

Prove the identity. $$\frac{\sin 3 x+\sin 7 x}{\cos 3 x-\cos 7 x}=\cot 2 x$$

Verify the identity. $$\frac{\tan v-\cot v}{\tan ^{2} v-\cot ^{2} v}=\sin v \cos v$$

Prove the identity. $$\frac{\sin 10 x}{\sin 9 x+\sin x}=\frac{\cos 5 x}{\cos 4 x}$$

Verify the identity. $$\frac{1+\sin x}{1-\sin x}=(\tan x+\sec x)^{2}$$

Prove the identity. $$\frac{\sin x+\sin 3 x+\sin 5 x}{\cos x+\cos 3 x+\cos 5 x}=\tan 3 x$$

Verify the identity. $$\frac{\tan x+\tan y}{\cot x+\cot y}=\tan x \tan y$$

Prove the identity. $$\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \left(\frac{x+y}{2}\right)$$

Verify the identity. $$(\tan x+\cot x)^{4}=\csc ^{4} x \sec ^{4} x$$

Prove the identity. $$\tan y=\frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}$$

Verify the identity. $$(\sin \alpha-\tan \alpha)(\cos \alpha-\cot \alpha)=(\cos \alpha-1)(\sin \alpha-1)$$

Show that \(\sin 130^{\circ}-\sin 110^{\circ}=-\sin 10^{\circ}\)

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that \(0 \leq \theta<\pi / 2\) $$\frac{x}{\sqrt{1-x^{2}}}, \quad x=\sin \theta$$

Show that \(\cos 100^{\circ}-\cos 200^{\circ}=\sin 50^{\circ}\)

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that \(0 \leq \theta<\pi / 2\) $$\sqrt{1+x^{2}}, \quad x=\tan \theta$$

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that \(0 \leq \theta<\pi / 2\) $$\sqrt{x^{2}-1}, \quad x=\sec \theta$$