Chapter 6

Find the exact value of the expression whenever It is defined. (a) \(\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) (b) \(\cos ^{-1}\left(-\frac{1}{2}\right)\) (c) \(\tan ^{-1}(-\sqrt{3})\)

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

Find the exact values of \(\sin 2 \theta, \cos 2 \theta,\) and \(\tan 2 \theta\) for the given values of \(\theta.\) $$\cos \theta=\frac{3}{5} ; \quad 0^{\circ}<\theta<90^{\circ}$$

Exer. 1-4: Express as a cofunction of a complementary angle. (a) \(\sin 46^{\circ} 37^{\prime}\) (b) \(\cos 73^{\circ} 12^{\prime}\) (c) \(\tan \frac{\pi}{6}\) (d) \(\sec 17.28^{\circ}\)

Verify the Identity. $$\csc \theta-\sin \theta=\cot \theta \cos \theta$$

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

Find the exact value of the expression whenever It is defined. (a) \(\sin ^{-1}\left(-\frac{1}{2}\right)\) (b) \(\cos ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) (c) \(\tan ^{-1}(-1)\)

Express as a sum or difference. $$\sin (-4 x) \cos 8 x$$

Find the exact values of \(\sin 2 \theta, \cos 2 \theta,\) and \(\tan 2 \theta\) for the given values of \(\theta.\) $$\text { cot } \theta=\frac{4}{3} ; \quad 180^{\circ}<\theta<270^{\circ}$$

Exer. 1-4: Express as a cofunction of a complementary angle. (a) \(\tan 24^{\circ} 12^{\prime}\) (b) \(\sin 89^{\circ} 41^{\prime}\) (c) \(\cos \frac{\pi}{3}\) d) \(\cot 61.87^{\circ}\)

Verify the Identity. $$\sin x+\cos x \cot x=\csc x$$

Find all solutions of the equation. $$\cos t=-1$$

Find the exact value of the expression whenever It is defined. (a) \(\arcsin \frac{\sqrt{3}}{2}\) (b) \(\arccos \frac{\sqrt{2}}{2}\) (c) \(\arctan \frac{1}{\sqrt{3}}\)

Express as a sum or difference. $$\cos 6 u \cos (-4 u)$$

Find the exact values of \(\sin 2 \theta, \cos 2 \theta,\) and \(\tan 2 \theta\) for the given values of \(\theta.\) $$\sec \theta=-3 ; \quad 90^{\circ}<\theta<180^{\circ}$$

Exer. 1-4: Express as a cofunction of a complementary angle. (a) \(\cos \frac{7 \pi}{20}\) \(\sin \frac{1}{4}\) (c) \(\tan 1\) (d) \(\csc 0.53\)

Verify the Identity. $$\frac{\sec ^{2} 2 u-1}{\sec ^{2} 2 u}=\sin ^{2} 2 u$$

Find all solutions of the equation. $$\tan \theta=\sqrt{3}$$

Find the exact value of the expression whenever It is defined. (a) arcsin 0 (b) arccos \((-1)\) (c) arctan 0

Express as a sum or difference. $$\cos 4 t \sin 6 t$$

Find the exact values of \(\sin 2 \theta, \cos 2 \theta,\) and \(\tan 2 \theta\) for the given values of \(\theta.\) $$\sin \theta=-\frac{4}{5} ; \quad 270^{\circ}<\theta<360^{\circ}$$

Exer. 1-4: Express as a cofunction of a complementary angle. (a) \(\sin \frac{\pi}{12}\) (b) \(\cos 0.64\) (c) \(\tan \sqrt{2}\) d) \(\sec 1.2\)

Verify the Identity. $$\tan t+2 \cos t \csc t=\sec t \csc t+\cot t$$

Find all solutions of the equation. $$\cot \alpha=-\frac{1}{\sqrt{3}}$$

Find the exact value of the expression whenever It is defined. (a) \(\sin ^{-1} \frac{\pi}{3}\) (b) \(\cos ^{-1} \frac{\pi}{2}\) \((c) \tan ^{-1} 1\)

Express as a sum or difference. $$2 \sin 9 \theta \cos 3 \theta$$

Find the exact values of \(\sin (\theta / 2), \cos (\theta / 2),\) and \(\tan (\theta / 2)\) for the given conditions. $$\sec \theta=\frac{5}{4} ; \quad 0^{\circ}<\theta<90^{\circ}$$

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

Verify the Identity. $$\frac{\csc ^{2} \theta}{1+\tan ^{2} \theta}=\cot ^{2} \theta$$

Find all solutions of the equation. $$\sec \beta=2$$

Express as a sum or difference. $$2 \sin 7 \theta \sin 5 \theta$$

Find the exact values of \(\sin (\theta / 2), \cos (\theta / 2),\) and \(\tan (\theta / 2)\) for the given conditions. $$\csc \theta=-\frac{5}{3} ; \quad-90^{\circ}<\theta<0^{\circ}$$

Exer. \(5-10:\) Find the exact values. (a) \(\sin \frac{2 \pi}{3}+\sin \frac{\pi}{4}\) b) \(\sin \frac{11 \pi}{12}\left(\text { use } \frac{11 \pi}{12}=\frac{2 \pi}{3}+\frac{\pi}{4}\right)\)

Verify the Identity. $$(\tan u+\cot u)(\cos u+\sin u)=\csc u+\sec u$$

Find all solutions of the equation. $$\csc \gamma=\sqrt{2}$$

Find the exact value of the expression whenever It is defined. (a) \(\sin \left[\arcsin \left(-\frac{3}{10}\right)\right]\) (b) \(\cos \left(\arccos \frac{1}{2}\right)\) (c) \(\tan (\arctan 14)\)

Express as a sum or difference. $$3 \cos x \sin 2 x$$

Find the exact values of \(\sin (\theta / 2), \cos (\theta / 2),\) and \(\tan (\theta / 2)\) for the given conditions. $$\tan \theta=1 ; \quad-180^{\circ}<\theta<-90^{\circ}$$

Exer. \(5-10:\) Find the exact values. (a) \(\tan 60^{\circ}+\tan 225^{\circ}\) (b) \(\left.\tan 285^{\circ} \text { (use } 285^{\circ}=60^{\circ}+225^{\circ}\right)\)

Verify the Identity. $$\frac{1+\cos 3 t}{\sin 3 t}+\frac{\sin 3 t}{1+\cos 3 t}=2 \csc 3 t$$

Find the exact value of the expression whenever It is defined. (a) \(\sin \left(\sin ^{-1} \frac{2}{3}\right)\) (b) \(\cos \left[\cos ^{-1}\left(-\frac{1}{5}\right)\right]\) (c) \(\tan \left[\tan ^{-1}(-9)\right]\)

Express as a sum or difference. $$5 \cos u \cos 5 u$$

Find the exact values of \(\sin (\theta / 2), \cos (\theta / 2),\) and \(\tan (\theta / 2)\) for the given conditions. $$\sec \theta=-4 ; \quad 180^{\circ}<\theta<270^{\circ}$$

Exer. \(5-10:\) Find the exact values. (a) \(\cos 135^{\circ}-\cos 60^{\circ}\) (b) \(\left.\cos 75^{\circ} \text { (use } 75^{\circ}=135^{\circ}-60^{\circ}\right)\)

Verify the Identity. $$\tan ^{2} \alpha-\sin ^{2} \alpha=\tan ^{2} \alpha \sin ^{2} \alpha$$

Find the exact value of the expression whenever It is defined. \(\sin ^{-1}\left(\sin \frac{\pi}{3}\right)\) (b) \(\cos ^{-1}\left[\cos \left(\frac{5 \pi}{6}\right)\right]\) (c) \(\tan ^{-1}\left[\tan \left(-\frac{\pi}{6}\right)\right]\)

Express as a product. $$\sin 6 \theta+\sin 2 \theta$$

Use half-angle formulas to find the exact values. (a) \(\cos 67^{\circ} 30^{\prime}\) (b) \(\sin 15^{\circ}\) (c) \(\tan \frac{3 \pi}{8}\)

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

Verify the Identity. $$\frac{1}{1-\cos \gamma}+\frac{1}{1+\cos \gamma}=2 \csc ^{2} \gamma$$