Chapter 7

Show that the restricted secant function, whose domain consists of all numbers \(x\) such that \(0 \leq x \leq \pi\) and \(x \neq \pi / 2,\) has an inverse function. Sketch its graph.

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$\sin ^{2} x+2 \sin x-2=0$$

Show that the restricted cosecant function, whose domain consists of all numbers \(x\) such that \(-\pi / 2 \leq x \leq \pi / 2\) and \(x \neq 0,\) has an inverse function. Sketch its graph.

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

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$\cos ^{2} x+5 \cos x=1$$

Show that the restricted cotangent function, whose domain is the interval \((0, \pi),\) has an inverse function. Sketch its graph.

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

Prove the identity. $$\log _{10}(\cot x)=-\log _{10}(\tan x)$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$4 \cos ^{2} x-2 \cos x=1$$

Prove the identity. $$\log _{10}(\sec x)=-\log _{10}(\cos x)$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$2 \tan ^{2} x-1=3 \tan x$$

Prove the identity. $$\log _{10}(\csc x+\cot x)=-\log _{10}(\csc x-\cot x)$$

Prove the identity. \(\tan ^{-1}(-x)=-\tan ^{-1} x\)

Prove the identity. $$\log _{10}(\sec x+\tan x)=-\log _{10}(\sec x-\tan x)$$

Prove the identity. \(\cos ^{-1}(-x)=\pi-\cos ^{-1} x\) [Hint: Let \(u=\cos ^{-1}(-x)\) and show that \(0 \leq \pi-u \leq \pi ;\) use the identity \(\cos (\pi-u)=-\cos u .]\)

Prove the identity. $$\tan x-\tan y=-\tan x \tan y(\cot x-\cot y)$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$\sec ^{2} x-2 \tan ^{2} x=0$$

Prove the identity. \(\sin ^{-1}(\cos x)=\pi / 2-x \quad(0 \leq x \leq \pi)\)

Prove the identity. $$\frac{\tan x-\tan y}{\cot x-\cot y}=-\tan x \tan y$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$\sin 2 x+\cos x=0$$

Prove the identity. \(\tan ^{-1}(\cot x)=\pi / 2-x \quad(0

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

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

Prove the identity. \(\sin ^{-1} x=\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right) \quad(-1

Prove the identity. \(\cos ^{-1} x=\frac{\pi}{2}-\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right) \quad(-1

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$\cos ^{2} x-\sin ^{2} x+\sin x=0$$

Is it true that \(\tan ^{-1} x=\frac{\sin ^{-1} x}{\cos ^{-1} x}\) ? Justify your answer.

Prove the identity. $$\frac{\cos 8 x+\cos 4 x}{\cos 8 x-\cos 4 x}=-\cot 6 x \cot 2 x$$

Using the viewing window with \(-2 \pi \leq x \leq 2 \pi\) and \(-4 \leq y \leq 4\) graph the functions \(f(x)=\cos \left(\cos ^{-1} x\right)\) and \(g(x)=\cos ^{-1}(\cos x) .\) How do you explain the shapes of the two graphs?

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

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

Prove the identity. (a) Prove that \(\frac{1-\cos x}{\sin x}=\frac{\sin x}{1+\cos x}\) (b) Use part (a) and the half-angle identity proved in the text to prove that $$ \tan \frac{x}{2}=\frac{\sin x}{1+\cos x} $$

Solve the equation graphically. $$5 \sin 3 x+6 \cos 3 x=1$$

Prove the identity. (a) List the exact values of \(\cos \frac{\pi}{4}, \cos \frac{\pi}{8}, \cos \frac{\pi}{16},\) and \(\cos \frac{\pi}{32} .[\text { Hint: Exercises } 11,23 \text { and } 91 .]\) (b) Based on the pattern you see in the answers to part (a) make a conjecture about the exact value of \(\cos \frac{\pi}{64}\) Use a calculator to support your answer. (c) Make a conjecture about the exact value of \(\cos \frac{\pi}{128}\) and support the truth of your conjecture with a calculator. (d) What do you think the exact value of \(\cos \frac{\pi}{256}\) is?

Solve the equation graphically. $$\sin ^{2} 2 x-3 \cos 2 x+2=0$$

Solve the equation graphically. $$\tan x=3 \cos x$$

Solve the equation graphically. $$\sin ^{3} x+2 \sin ^{2} x-3 \cos x+2=0$$

The number of hours of daylight in Detroit on day \(t\) of a non-leap year (with \(t=0\) being January 1 ) is given by the function $$d(t)=3 \sin \left[\frac{2 \pi}{365}(t-80)\right]+12$$ (a) On what days of the year are there exactly 11 hours of daylight? (b) What day has the maximum amount of daylight?

Under what conditions (on the constant) does a basic equation involving the sine and cosine function have \(n o\) solutions?

Let \(n\) be a fixed positive integer. Describe all solutions of the equation \(\sin n x=1 / 2 . \text { [Hint: See Exercises } 43-52 .]\)