Chapter 6

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\csc u=3, \quad 0

Find the exact values of the sine, cosine, and tangent of the angle. $$-\frac{23 \pi}{12}$$

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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\sec x-\cos x, \quad y_{2}=\sin x \tan x$$

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

Use a double-angle formula to rewrite the expression. Use a graphing utility to graph both expressions to verify that both forms are the same. $$6 \sin x \cos x$$

Write the expression as the sine, cosine, or tangent of an angle. $$\sin 60^{\circ} \cos 10^{\circ}-\cos 60^{\circ} \sin 10^{\circ}$$

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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\sin x+\cos x \cot x, \quad y_{2}=\csc x$$

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

Use a double-angle formula to rewrite the expression. Use a graphing utility to graph both expressions to verify that both forms are the same. $$14 \sin x \cos x$$

Write the expression as the sine, cosine, or tangent of an angle. $$\sin 110^{\circ} \cos 80^{\circ}+\cos 110^{\circ} \sin 80^{\circ}$$

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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\cos x+\sin x \tan x, \quad y_{2}=\sec x$$

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

Use a double-angle formula to rewrite the expression. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{2} x-\frac{1}{2}$$

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 325^{\circ}-\tan 116^{\circ}}{1+\tan 325^{\circ} \tan 116^{\circ}}$$

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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\frac{1}{\tan x}+\frac{1}{\cot x} ; \quad y_{2}=\tan x+\cot x$$

Solve the equation. $$2 \sin x+1=0$$

Use a double-angle formula to rewrite the expression. Use a graphing utility to graph both expressions to verify that both forms are the same. $$10 \sin ^{2} x-5$$

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 154^{\circ}-\tan 49^{\circ}}{1+\tan 154^{\circ} \tan 49^{\circ}}$$

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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\frac{1}{\sin x}-\frac{1}{\csc x}, \quad y_{2}=\csc x-\sin x$$

Solve the equation. $$\sqrt{2} \sin x+1=0$$

Rewrite the function using the power-reducing formulas. Then use a graphing utility to graph the function. $$f(x)=\sin ^{2} x$$

Write the expression as the sine, cosine, or tangent of an angle. $$\cos \frac{\pi}{9} \cos \frac{\pi}{7}-\sin \frac{\pi}{9} \sin \frac{\pi}{7}$$

Describe the error. $$\begin{aligned} &(1+\tan x)[1+\cot (-x)]\\\ &\begin{array}{l} =(1+\tan x)(1+\cot x) \\ =1+\cot x+\tan x+\tan x \cot x \\ =1+\cot x+\tan x+1 \\ =2+\cot x+\tan x \end{array} \end{aligned}$$

Solve the equation. $$\sqrt{3} \csc x-2=0$$

Rewrite the function using the power-reducing formulas. Then use a graphing utility to graph the function. $$f(x)=\cos ^{2} x$$

Write the expression as the sine, cosine, or tangent of an angle. $$\sin \frac{4 \pi}{9} \cos \frac{\pi}{8}+\cos \frac{4 \pi}{9} \sin \frac{\pi}{8}$$

Solve the equation. $$\cot x+1=0$$

Rewrite the function using the power-reducing formulas. Then use a graphing utility to graph the function. $$f(x)=\cos ^{3} x$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\cot x \sin x$$

Write the expression as the sine, cosine, or tangent of an angle. $$\sin 3.5 x \cos 1.2 y+\cos 3.5 x \sin 1.2 y$$

Fill in the missing step(s). $$\begin{aligned} \sec ^{4} x-2 \sec ^{2} x+1 &=\left(\sec ^{2} x-1\right)^{2} \\ &= \text { _____ } \\ &=\tan ^{4} x \end{aligned}$$

Solve the equation. $$3 \sec ^{2} x-4=0$$

Rewrite the function using the power-reducing formulas. Then use a graphing utility to graph the function. $$f(x)=\sin ^{3} x$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\cos \beta \tan \beta$$

Fill in the missing step(s). $$\begin{aligned} \frac{\tan x-\cot x}{\tan x+\cot x} &=\frac{\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}}{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}} \\ &= \text { _____} \\ &=\frac{\sin ^{2} x-\cos ^{2} x}{1} \\ &=\sin ^{2} x-\cos ^{2} x \\ &= \text { _____} \\ &=1-2 \cos ^{2} x \end{aligned}$$

Solve the equation. $$3 \cot ^{2} x-1=0$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{4} x$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\sin \phi(\csc \phi-\sin \phi)$$

Find the exact value of the expression. $$\sin \frac{\pi}{12} \cos \frac{\pi}{4}+\cos \frac{\pi}{12} \sin \frac{\pi}{4}$$

Verify the identity. $$\cot \left(\frac{\pi}{2}-x\right) \csc x=\sec x$$

Solve the equation. $$\sin x(\sin x+1)=0$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\sin ^{2} x \cos ^{2} x$$

Find the exact value of the expression. $$\cos \frac{\pi}{16} \cos \frac{3 \pi}{16}-\sin \frac{\pi}{16} \sin \frac{3 \pi}{16}$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\cos x\left(1+\tan ^{2} x\right)$$

Verify the identity. $$\frac{\cos \left[\left(\frac{\pi}{2}\right)-x\right]}{\sin \left[\left(\frac{\pi}{2}\right)-x\right]}=\tan x$$

Solve the equation. $$\cos x(\cos x-1)=0$$

Find the exact value of the expression. $$\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ}$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{\csc x}{\cot x}$$

Verify the identity. $$\frac{\csc (-x)}{\sec (-x)}=-\cot x$$

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$\cos x+1=-\cos x$$