Chapter 6

Use the values to evaluate (if possible) all six trigonometric functions. $$\cos \left(\frac{\pi}{2}-x\right)=\frac{3}{5}, \quad \cos x=\frac{4}{5}$$

Verify the identity. $$\frac{\sin ^{2} t}{\tan ^{2} t}=\cos ^{2} t$$

Solving a Trigonometric Equation In Exercises \(11-16\) fF\(\left[0^{\circ}, 360^{\circ}\right)\). $$\cos x=\frac{\sqrt{3}}{2}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\cos 2 x-\cos x=0$$

Use the values to evaluate (if possible) all six trigonometric functions. $$\sin (-x)=-\frac{2}{3}, \quad \tan x=-\frac{2 \sqrt{5}}{5}$$

Verify the identity. $$\cos ^{2} \beta-\sin ^{2} \beta=1-2 \sin ^{2} \beta$$

Solving a Trigonometric Equation In Exercises \(11-16\) fF\(\left[0^{\circ}, 360^{\circ}\right)\). $$\tan x=1$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\cos 2 x+\sin x=0$$

Use the values to evaluate (if possible) all six trigonometric functions. $$\csc (-x)=-5, \quad \cos x=\frac{\sqrt{24}}{5}$$

Verify the identity. $$\cot ^{2} \beta+\csc ^{2} \beta=2 \csc ^{2} \beta-1$$

Solving a Trigonometric Equation In Exercises \(11-16\) fF\(\left[0^{\circ}, 360^{\circ}\right)\). $$\tan x=-\sqrt{3}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\sin 4 x=-2 \sin 2 x$$

Use the values to evaluate (if possible) all six trigonometric functions. $$\tan \theta=2, \quad \sin \theta<0$$

Verify the identity. $$\tan ^{2} \theta+6=\sec ^{2} \theta+5$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos x=-\frac{\sqrt{3}}{2}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$(\sin 2 x+\cos 2 x)^{2}=1$$

Use the values to evaluate (if possible) all six trigonometric functions. $$\sec \theta=4, \quad \tan \theta<0$$

Verify the identity. $$3+\sin ^{2} z=4-\cos ^{2} z$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin x=-\frac{1}{2}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\tan 2 x-\cot x=0$$

Find the exact values of the sine, cosine, and tangent of the angle. $$285^{\circ}$$

Use the values to evaluate (if possible) all six trigonometric functions. $$\csc \theta \text { is undefined, } \cos \theta<0$$

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

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot x=-1$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\tan 2 x-2 \cos x=0$$

Find the exact values of the sine, cosine, and tangent of the angle. $$15^{\circ}$$

Use the values to evaluate (if possible) all six trigonometric functions. $$\tan \theta \text { is undefined, } \sin \theta>0$$

Verify the identity. $$\tan ^{2} y\left(\csc ^{2} y-1\right)=1$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin x=\frac{\sqrt{3}}{2}$$

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\sin u=-\frac{3}{5}, \quad 3 \pi / 2

Find the exact values of the sine, cosine, and tangent of the angle. $$-105^{\circ}$$

Use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\frac{\sec \theta-1}{1-\cos \theta}, \quad y_{2}=\sec \theta$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\tan x=-\frac{\sqrt{3}}{3}$$

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\cos u=-\frac{2}{3}, \quad \pi / 2

Find the exact values of the sine, cosine, and tangent of the angle. $$-165^{\circ}$$

Match the trigonometric expression with its simplified form. (a) \(\sec x\) (b) -1 (c) \(\cot x\) (d) 1 (e) \(-\tan x\) (f) \(\sin x\) $$\tan x \csc x$$

Use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\frac{\csc x-1}{1-\sin x}, \quad y_{2}=\csc x$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos x=\frac{\sqrt{2}}{2}$$

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\tan u=\frac{1}{2}, \quad \pi

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{13 \pi}{12}$$

Use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\left(1+\cot ^{2} x\right) \cos ^{2} x, \quad y_{2}=\cot ^{2} x$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\csc x=-2$$

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\cot u=-6, \quad 3 \pi / 2

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{5 \pi}{12}$$

Use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\left(1+\tan ^{2} x\right) \sin ^{2} x, \quad y_{2}=\tan ^{2} x$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec x=\sqrt{2}$$

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\sec u=-2, \quad \pi / 2

Find the exact values of the sine, cosine, and tangent of the angle. $$-\frac{7 \pi}{12}$$

Use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\csc x-\sin x, \quad y_{2}=\cos x \cot x$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot x=\sqrt{3}$$