Chapter 10

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$\tan \theta+\cot \theta=0$$

Write expression as a single trigonometric function or a power of a trigonometric function. (You may wish to use a graph to support your result.) $$\frac{\sin ^{2} x}{\cos ^{2} x}+\sin x \csc x$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(135^{\circ}-x\right)$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin \theta-\sin 2 \theta=0$$

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\arcsin \left(-\frac{\sqrt{2}}{2}\right)$$

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\cos 67.5^{\circ}$$

Write expression as a single trigonometric function or a power of a trigonometric function. (You may wish to use a graph to support your result.) $$\frac{1}{\tan ^{2} \alpha}+\cot \alpha \tan \alpha$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\sin \left(45^{\circ}+x\right)$$

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\sin x+\sin 3 x=\cos x$$

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\cot ^{-1}\left(-\frac{\sqrt{3}}{3}\right)$$

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\sin 67.5^{\circ}$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$9 \sin ^{2} \theta-6 \sin \theta=1$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\cot \theta \sin \theta$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan \left(45^{\circ}+x\right)$$

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\sin 3 x-\sin x=0$$

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\sec ^{-1}(-2)$$

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\tan 195^{\circ}$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$4 \cos ^{2} \theta+4 \cos \theta=1$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\sec \theta \cot \theta \sin \theta$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan (\pi+x)$$

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\cos 2 x+\cos x=0$$

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\csc ^{-1}(-2)$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\cos \frac{x}{2}, \text { given } \cos x=\frac{1}{4} \text { and } 0 < x < \frac{\pi}{2}$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$\tan ^{2} \theta+4 \tan \theta+2=0$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\cos \theta \csc \theta$$

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\sin 4 x+\sin 2 x=2 \cos x$$

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\csc ^{-1}(-1)$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\sin \frac{x}{2}, \text { given } \cos x=-\frac{5}{8} \text { and } \frac{\pi}{2} < x < \pi$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$3 \cot ^{2} \theta-3 \cot \theta-1=0$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{\sec \theta}{\csc \theta}$$

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\cos \frac{x}{2}=2 \sin 2 x$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\tan \frac{x}{2}, \text { given } \sin x=\frac{3}{5} \text { and } \frac{\pi}{2} < x < \pi$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$\sin ^{2} \theta-2 \sin \theta+3=0$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{\cot ^{2} \theta}{\csc ^{2} \theta}$$

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\sin \frac{x}{2}+\cos 3 x=0$$

Use identities to write each expression as a finction with \(x\) as the only argument. $$\tan \left(x+\frac{7 \pi}{4}\right)$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\cos \frac{x}{2}, \text { given } \sin x=-\frac{4}{5} \text { and } \frac{3 \pi}{2} < x < 2 \pi$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$2 \cos ^{2} \theta+2 \cos \theta+1=0$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{\tan ^{2} \theta}{\sec ^{2} \theta}$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\tan \frac{x}{2}, \text { given } \tan x=\frac{\sqrt{7}}{3} \text { and } \pi < x < \frac{3 \pi}{2}$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$\cot \theta+2 \csc \theta=3$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$1-\cos ^{2} \theta$$

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\csc ^{-1} \frac{1}{3}$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\tan \frac{x}{2}, \text { given } \tan x=-\frac{\sqrt{5}}{2} \text { and } \frac{\pi}{2} < x < \pi$$

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$2 \sin \theta=1-2 \cos \theta$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{1}{1+\cot ^{2} \theta}$$

Suppose that A and B are angles in standand position. Use the given information to find (a) \(\sin (A+B)\), (b) \(\sin (A-B)\), (c) \(\tan (A+B)\), (d) \(\tan (A-B)\), (e) the quadrant of \(A+B\), and ( \(f\) ) the quadrant of \(A-B\). Do not use a calculator. $$\cos A=\frac{3}{5}, \sin B=\frac{5}{13}, \quad 0

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\sin ^{-1}(-0.13349122)$$

$$\tan \frac{x}{2}, \text { given } \tan x=-\frac{\sqrt{5}}{2} \text { and } \frac{\pi}{2}

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$8 \cos ^{2} \theta+18 \cos \theta=5$$