Chapter 10

Write short answers and fill in the blanks. Consider the three other inverse trigonometric functions, as defined in this section. (a) Give the domain and range of the inverse cosecant function. (b) Give the domain and range of the inverse secant function. (c) Give the domain and range of the inverse cotangent function.

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \sec x+1=\sec x+3$$

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\sin (-2.5)$$

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos \left(\frac{13 \pi}{12}\right)$$

Solve each equation over the interval \([0,2 \pi)\) $$\sin 2 x-\cos x=0$$

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\tan ^{-1} 1$$

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\sin \theta=\frac{4}{5} \text { and } \cos \theta<0$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$(\cot x-1)(\sqrt{3} \cot x+1)=0$$

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\tan \left(-\frac{\pi}{7}\right)$$

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos 75^{\circ}$$

How many solutions does \(\sin x=\frac{1}{2}\) have on \([0,2 \pi) ?\) How many solutions does \(\sin 2 x=\frac{1}{2}\) have on \([0,2 \pi)\) ? Explain how a graph supports your answer.

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\sin ^{-1} 0$$

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\cos \theta=-\frac{12}{13} \text { and } \sin \theta>0$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$(\csc x+2)(\csc x-\sqrt{2})=0$$

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\cot \left(-\frac{4 \pi}{7}\right)$$

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin 105^{\circ}$$

Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(\cos 2 x=\frac{\sqrt{3}}{2}\) (b) \(\cos 2 x>\frac{\sqrt{3}}{2}\)

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\cos ^{-1}(-1)$$

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\tan \theta=2 \text { and } \cos \theta>0$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$\cos x \cot x=\cos x$$

For expression in Column I, choose the expression from Column II that completes a fundamental identity. Do not use a calculator. \(\mathbf{I}\) \(\frac{\cos x}{\sin x}=\)_______ \(\mathbf{II}\) A. \(\sin ^{2} x+\cos ^{2} x\) B. cot \(x\) C. \(\sec ^{2} x\) D. \(\frac{\sin x}{\cos x}\) E. \(\cos x\)

Use identities to find the exact value of each expression. Do not use a calculator. $$\tan 105^{\circ}$$

Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(\cos 2 x=-\frac{1}{2}\) (b) \(\cos 2 x>-\frac{1}{2}\)

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\arctan (-1)$$

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\tan \theta=\frac{5}{3} \text { and } \sin \theta<0$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$\sin x \cot x=\sin x$$

For expression in Column I, choose the expression from Column II that completes a fundamental identity. Do not use a calculator. \(\mathbf{I}\) \(\tan x=\)_______ \(\mathbf{II}\) A. \(\sin ^{2} x+\cos ^{2} x\) B. cot \(x\) C. \(\sec ^{2} x\) D. \(\frac{\sin x}{\cos x}\) E. \(\cos x\)

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin \left(-15^{\circ}\right)$$

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\sin ^{-1}(-1)$$

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\sin \theta=-\frac{\sqrt{5}}{7} \text { and } \cos \theta>0$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$\sin ^{2} x-2 \sin x+1=0$$

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos \left(-15^{\circ}\right)$$

For expression in Column I, choose the expression from Column II that completes a fundamental identity. Do not use a calculator. \(\mathbf{I}\) \(\cos (-x)=\) _______ \(\mathbf{II}\) A. \(\sin ^{2} x+\cos ^{2} x\) B. cot \(x\) C. \(\sec ^{2} x\) D. \(\frac{\sin x}{\cos x}\) E. \(\cos x\)

Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(\sin 3 x=0\) (b) \(\sin 3 x<0\)

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\cos ^{-1} \frac{1}{2}$$

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\cos \theta=\frac{\sqrt{3}}{5} \text { and } \sin \theta>0$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$\cos ^{2} x+2 \cos x+1=0$$

For expression in Column I, choose the expression from Column II that completes a fundamental identity. Do not use a calculator. \(\mathbf{I}\) \(\tan ^{2} x+1=\) _______ \(\mathbf{II}\) A. \(\sin ^{2} x+\cos ^{2} x\) B. cot \(x\) C. \(\sec ^{2} x\) D. \(\frac{\sin x}{\cos x}\) E. \(\cos x\)

Use identities to find the exact value of each expression. Do not use a calculator. $$\tan \left(-75^{\circ}\right)$$

Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(\sqrt{2} \cos 2 x=-1\) (b) \(\sqrt{2} \cos 2 x \leq-1\)

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\arctan 0$$

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$\cos ^{2} 15^{\circ}-\sin ^{2} 15^{\circ}$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$4(1+\sin x)(1-\sin x)=3$$

For expression in Column I, choose the expression from Column II that completes a fundamental identity. Do not use a calculator. \(\mathbf{I}\) \(1=\) _______ \(\mathbf{II}\) A. \(\sin ^{2} x+\cos ^{2} x\) B. cot \(x\) C. \(\sec ^{2} x\) D. \(\frac{\sin x}{\cos x}\) E. \(\cos x\)

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos \frac{\pi}{3} \cos \frac{2 \pi}{3}-\sin \frac{\pi}{3} \sin \frac{2 \pi}{3}$$

Solve each equation in part (a) analytically over the interval \([0,2 \pi) .\) Then use a graph to solve each inequality in part (b). (a) \(2 \sqrt{3} \sin 2 x=\sqrt{3}\) (b) \(2 \sqrt{3} \sin 2 x \leq \sqrt{3}\)

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\arcsin \left(-\frac{\sqrt{3}}{2}\right)$$

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$\frac{2 \tan 15^{\circ}}{1-\tan ^{2} 15^{\circ}}$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$(\cot x-\sqrt{3})(2 \sin x+\sqrt{3})=0$$

For e expression in Column I, choose the expression from Column II that completes an identity. You may have to rewrite one or both expressions. Do not use a calculator. \(\mathbf{I}\) \(-\tan x \cos x=\) _______ \(\mathbf{II}\) A. \(\frac{\sin ^{2} x}{\cos ^{2} x}\) B. \(\frac{1}{\sec ^{2} x}\) C. \(\sin (-x)\) D. \(\csc ^{2} x-\cot ^{2} x+\sin ^{2} x\) E. \(\tan x\)