Chapter 7

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\arcsin \left[\sin \left(-\frac{\pi}{2}\right)\right]\) (b) \(\arccos (\cos 0)\) (c) \(\arctan \left(\tan \frac{\pi}{4}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ \csc \theta \sin \theta=1 $$

Exer. 1-50: Verify the identity. $$ \frac{1+\csc 3 \beta}{\sec 3 \beta}-\cot 3 \beta=\cos 3 \beta $$

Express as a sum or difference. $$ \sin 4 \theta-\sin 8 \theta $$

Exer. 5-10: Find the exact values. (a) \(\tan \frac{3 \pi}{4}-\tan \frac{\pi}{6}\) (b) \(\tan \frac{7 \pi}{12}\left(\right.\) use \(\left.\frac{7 \pi}{12}=\frac{3 \pi}{4}-\frac{\pi}{6}\right)\)

Use half-angle formulas to find the exact values. (a) \(\cos 165^{\circ}\) (b) \(\sin 157^{\circ} 30^{\prime}\) (c) \(\tan \frac{\pi}{8}\)

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\arcsin \left(\sin \frac{5 \pi}{4}\right)\) (b) \(\arccos \left(\cos \frac{5 \pi}{4}\right)\) (c) \(\arctan \left(\tan \frac{7 \pi}{4}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ 2 \cos 2 \theta-\sqrt{3}=0 $$

Exer. 1-50: Verify the identity. $$ (\sec u-\tan u)(\csc u+1)=\cot u $$

Express as a sum or difference. $$ \cos 5 x-\cos 3 x $$

Exer. 11-16: Express as a trigonometric function of one angle. $$ \cos 48^{\circ} \cos 23^{\circ}+\sin 48^{\circ} \sin 23^{\circ} $$

Verify the identity. $$ \sin 10 \theta=2 \sin 5 \theta \cos 5 \theta $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)\) (b) \(\cos ^{-1}\left(\cos \frac{4 \pi}{3}\right)\) (c) \(\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ 2 \sin 3 \theta+\sqrt{2}=0 $$

Exer. 1-50: Verify the identity. $$ \frac{\cot \theta-\tan \theta}{\sin \theta+\cos \theta}=\csc \theta-\sec \theta $$

Express as a sum or difference. $$ \cos 5 t+\cos 6 t $$

Exer. 11-16: Express as a trigonometric function of one angle. $$ \cos 13^{\circ} \cos 50^{\circ}-\sin 13^{\circ} \sin 50^{\circ} $$

Verify the identity. $$ \cos ^{2} 3 x-\sin ^{2} 3 x=\cos 6 x $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[\cos ^{-1}\left(-\frac{1}{2}\right)\right]\) (b) \(\cos \left(\tan ^{-1} 1\right)\) (c) \(\tan \left[\sin ^{-1}(-1)\right]\)

Exer. 1-38: Find all solutions of the equation. $$ \sqrt{3} \tan \frac{1}{3} t=1 $$

Exer. 1-50: Verify the identity. $$ \csc ^{4} t-\cot ^{4} t=\csc ^{2} t+\cot ^{2} t $$

Express as a sum or difference. $$ \sin 3 t-\sin 7 t $$

Exer. 11-16: Express as a trigonometric function of one angle. $$ \cos 10^{\circ} \sin 5^{\circ}-\sin 10^{\circ} \cos 5^{\circ} $$

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

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left(\tan ^{-1} \sqrt{3}\right)\) (b) \(\cos \left(\sin ^{-1} 1\right)\) (c) \(\tan \left(\cos ^{-1} 0\right)\)

Exer. 1-38: Find all solutions of the equation. $$ \cos \frac{1}{4} x=-\frac{\sqrt{2}}{2} $$

Exer. 1-50: Verify the identity. $$ \cos ^{4} 2 \theta+\sin ^{2} 2 \theta=\cos ^{2} 2 \theta+\sin ^{4} 2 \theta $$

Express as a sum or difference. $$ \cos \theta-\cos 5 \theta $$

Exer. 11-16: Express as a trigonometric function of one angle. $$ \sin 57^{\circ} \cos 4^{\circ}+\cos 57^{\circ} \sin 4^{\circ} $$

Verify the identity. $$ \frac{\sin ^{2} 2 \alpha}{\sin ^{2} \alpha}=4-4 \sin ^{2} \alpha $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\cot \left(\sin ^{-1} \frac{2}{3}\right)\) (b) \(\sec \left[\tan ^{-1}\left(-\frac{3}{5}\right)\right]\) (c) \(\csc \left[\cos ^{-1}\left(-\frac{1}{4}\right)\right]\)

Exer. 1-38: Find all solutions of the equation. $$ \sin \left(\theta+\frac{\pi}{4}\right)=\frac{1}{2} $$

Exer. 1-50: Verify the identity. $$ \frac{\cos \beta}{1-\sin \beta}=\sec \beta+\tan \beta $$

Express as a sum or difference. $$ \cos x+\cos 2 x $$

Exer. 11-16: Express as a trigonometric function of one angle. $$ \cos 3 \sin (-2)-\cos 2 \sin 3 $$

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

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\cot \left[\sin ^{-1}\left(-\frac{2}{5}\right)\right]\) (b) \(\sec \left(\tan ^{-1} \frac{7}{4}\right)\) (c) \(\csc \left(\cos ^{-1} \frac{1}{5}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ \cos \left(x-\frac{\pi}{3}\right)=-1 $$

Exer. 1-50: Verify the identity. $$ \frac{1}{\csc y-\cot y}=\csc y+\cot y $$

Express as a sum or difference. $$ \sin 8 t+\sin 2 t $$

Exer. 11-16: Express as a trigonometric function of one angle. $$ \sin (-5) \cos 2+\cos 5 \sin (-2) $$

Verify the identity. $$ \csc 2 u=\frac{1}{2} \csc u \sec u $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left(\arcsin \frac{1}{2}+\arccos 0\right)\) (b) \(\cos \left[\arctan \left(-\frac{3}{4}\right)-\arcsin \frac{4}{5}\right]\) (c) \(\tan \left(\arctan \frac{4}{3}+\arccos \frac{8}{17}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ \sin \left(2 x-\frac{\pi}{3}\right)=\frac{1}{2} $$

Exer. 1-50: Verify the identity. $$ \frac{\tan ^{2} x}{\sec x+1}=\frac{1-\cos x}{\cos x} $$

Verify the identity. $$ \frac{\sin 4 t+\sin 6 t}{\cos 4 t-\cos 6 t}=\cot t $$

If \(\sin \alpha=-\frac{5}{13}\) and \(\tan \alpha>0\), find the exact value of \(\sin \left(\alpha-\frac{\pi}{3}\right)\)

Verify the identity. $$ \sin 3 u=\sin u\left(3-4 \sin ^{2} u\right) $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[\sin ^{-1} \frac{5}{13}-\cos ^{-1}\left(-\frac{3}{5}\right)\right]\) (b) \(\cos \left(\sin ^{-1} \frac{4}{5}+\tan ^{-1} \frac{3}{4}\right)\) (c) \(\tan \left[\cos ^{-1} \frac{1}{2}-\sin ^{-1}\left(-\frac{1}{2}\right)\right]\)

Exer. 1-38: Find all solutions of the equation. $$ \cos \left(4 x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2} $$