Chapter 7

Find all solutions of the given equation. $$\cos \theta+1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$2 \sin \frac{\theta}{3}+\sqrt{3}=0$$

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

Simplify the trigonometric expression. $$\tan \theta+\cos (-\theta)+\tan (-\theta)$$

Use an appropriate Half-Angle Formula to find the exact value of the expression. $$\tan \frac{5 \pi}{12}$$

Find all solutions of the given equation. $$\sin \theta+1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$\sec \frac{\theta}{2}=\cos \frac{\theta}{2}$$

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

Simplify the trigonometric expression. $$\frac{\cos x}{\sec x+\tan x}$$

Use an appropriate Half-Angle Formula to find the exact value of the expression. $$\sin \frac{9 \pi}{8}$$

Find all solutions of the given equation. $$\sqrt{2} \sin \theta+1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$\sin 2 \theta=3 \cos 2 \theta$$

Prove the identity. $$\sin (x-\pi)=-\sin x$$

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. $$\frac{\cos x}{\sec x \sin x}=\csc x-\sin x$$

Use an appropriate Half-Angle Formula to find the exact value of the expression. $$\sin \frac{11 \pi}{12}$$

Find all solutions of the given equation. $$\sqrt{2} \cos \theta-1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$\csc 3 \theta=5 \sin 3 \theta$$

Prove the identity. $$\cos (x-\pi)=-\cos x$$

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. $$\frac{\tan y}{\csc y}=\sec y-\cos y$$

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(2 \sin 18^{\circ} \cos 18^{\circ}\) (b) \(2 \sin 3 \theta \cos 3 \theta\)

Find all solutions of the given equation. $$5 \sin \theta-1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$\sec \theta-\tan \theta=\cos \theta$$

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

Verify the identity. $$\frac{\sin \theta}{\tan \theta}=\cos \theta$$

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\frac{2 \tan 7^{\circ}}{1-\tan ^{2} 7^{\circ}}\) (b) \(\frac{2 \tan 7 \theta}{1-\tan ^{2} 7 \theta}\)

Find all solutions of the given equation. $$4 \cos \theta+1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$\tan 3 \theta+1=\sec 3 \theta$$

Verify the identity. $$\frac{\tan x}{\sec x}=\sin x$$

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\cos ^{2} 34^{\circ}-\sin ^{2} 34^{\circ}\) (b) \(\cos ^{2} 5 \theta-\sin ^{2} 5 \theta\)

Find all solutions of the given equation. $$3 \tan ^{2} \theta-1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$3 \tan ^{3} \theta-3 \tan ^{2} \theta-\tan \theta+1=0$$

Prove the identity. $$\cos \left(x+\frac{\pi}{6}\right)+\sin \left(x-\frac{\pi}{3}\right)=0$$

Verify the identity. $$\frac{\cos u \sec u}{\tan u}=\cot u$$

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2}\) (b) \(2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\)

Find all solutions of the given equation. $$\cot \theta+1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$4 \sin \theta \cos \theta+2 \sin \theta-2 \cos \theta-1=0$$

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

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

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\frac{\sin 8^{\circ}}{1+\cos 8^{\circ}}\) (b) \(\frac{1-\cos 4 \theta}{\sin 4 \theta}\)

Find all solutions of the given equation. $$2 \cos ^{2} \theta-1=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$2 \sin \theta \tan \theta-\tan \theta=1-2 \sin \theta$$

Prove the identity. $$\sin (x+y)-\sin (x-y)=2 \cos x \sin y$$

Verify the identity. $$\sin B+\cos B \cot B=\csc B$$

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\sqrt{\frac{1-\cos 30^{\circ}}{2}}\) (b) \(\sqrt{\frac{1-\cos 8 \theta}{2}}\)

Find all solutions of the given equation. $$4 \sin ^{2} \theta-3=0$$

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$\sec \theta \tan \theta-\cos \theta \cot \theta=\sin \theta$$

Prove the identity. $$\cos (x+y)+\cos (x-y)=2 \cos x \cos y$$

Verify the identity. $$\cos (-x)-\sin (-x)=\cos x+\sin x$$

Use the Addition Formula for sine to prove the Double-Angle Formula for sine.

Find all solutions of the given equation. $$\tan ^{2} \theta-4=0$$