Chapter 10

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos \frac{7 \pi}{8} \cos \frac{\pi}{8}+\sin \frac{7 \pi}{8} \sin \frac{\pi}{8}$$

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) \(\sin \frac{x}{2}=\sqrt{2}-\sin \frac{x}{2}\) (b) \(\sin \frac{x}{2}>\sqrt{2}-\sin \frac{x}{2}\)

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

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. $$1-2 \sin ^{2} 15^{\circ}$$

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

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. \(\mathbf{I}\) \(\sec ^{2} x-1=\) _______ \(\mathbf{II}\) 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. \(\tan x\)

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin 76^{\circ} \cos 31^{\circ}-\cos 76^{\circ} \sin 31^{\circ}$$

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) \(\sin x=\sin 2 x\) (b) \(\sin x>\sin 2 x\)

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\tan ^{-1}(-\sqrt{3})$$

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. $$1-2 \sin ^{2} 22.5^{\circ}$$

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

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. \(\mathbf{I}\) \(\frac{\sec x}{\csc x}=\) _______ \(\mathbf{II}\) 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. \(\tan x\)

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin 40^{\circ} \cos 50^{\circ}+\cos 40^{\circ} \sin 50^{\circ}$$

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) \(\cos x=\cos 2 x\) (b) \(\cos x<\cos 2 x\)

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

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. $$2 \cos ^{2} 67.5^{\circ}-1$$

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

Use identities to find the exact value of each expression. Do not use a calculator. $$\frac{\tan 80^{\circ}+\tan 55^{\circ}}{1-\tan 80^{\circ} \tan 55^{\circ}}$$

The equation \(\cot \frac{x}{2}-\csc \frac{x}{2}-1=0\) has solution set \(\varnothing\) over the interval \([0,2 \pi) .\) The solution set to the inequality cot \(\frac{x}{2}-\csc \frac{x}{2}-1>0\) over this interval is \(\varnothing .\) Does the graph of \(y=\cot \frac{x}{2}-\csc \frac{x}{2}-1\) lie above or below the \(x\) -axis over this interval?

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

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. $$\cos ^{2} \frac{\pi}{8}-\frac{1}{2}$$

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

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. \(\mathbf{I}\) \(\cos ^{2} x=\) _______ \(\mathbf{II}\) 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. \(\tan x\)

Use identities to find the exact value of each expression. Do not use a calculator. $$\frac{\tan 80^{\circ}-\tan \left(-55^{\circ}\right)}{1+\tan 80^{\circ} \tan \left(-55^{\circ}\right)}$$

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. $$\sqrt{2} \sin 3 x-1=0$$

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

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. $$\frac{\tan 51^{\circ}}{1-\tan ^{2} 51^{\circ}}$$

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

A student writes "1 \(+\cot ^{2}=\csc ^{2}\) " Comment on this student's work.

Use identities to write each expression as a function with \(x\) as the only argument. $$\sin \left(180^{\circ}-x\right)$$

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. $$-2 \cos 2 x=\sqrt{3}$$

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

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. $$\frac{\tan 34^{\circ}}{2\left(1-\tan ^{2} 34^{\circ}\right)}$$

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

A student makes this claim: "since \(\sin ^{2} \theta+\cos ^{2} \theta=1\) I should also be able to say that \(\sin \theta+\cos \theta=1\) if I take the square root of each side." Comment on this student's statement.

Use identities to write each expression as a function with \(x\) as the only argument. $$\sin \left(270^{\circ}+x\right)$$

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. $$\cos \frac{\theta}{2}=1$$

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

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. $$\frac{1}{4}-\frac{1}{2} \sin ^{2} 47.1^{\circ}$$

Solve \((\mathbf{a}) f(x)=0,(\mathbf{b}) f(x)>0,\) and \((\mathbf{c}) f(x)<0\) over the interval \([0,2 \pi)\) $$f(x)=-2 \cos x+1$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(180^{\circ}+x\right)$$

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. $$\sin \frac{\theta}{2}=1$$

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

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. $$\frac{1}{8} \sin 29.5^{\circ} \cos 29.5^{\circ}$$

Solve \((\mathbf{a}) f(x)=0,(\mathbf{b}) f(x)>0,\) and \((\mathbf{c}) f(x)<0\) over the interval \([0,2 \pi)\) $$f(x)=2 \sin x+1$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(270^{\circ}-x\right)$$

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. $$2 \sqrt{3} \sin \frac{x}{2}=3$$

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

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. $$4 \sin 15^{\circ} \cos 15^{\circ}$$

Solve \((\mathbf{a}) f(x)=0,(\mathbf{b}) f(x)>0,\) and \((\mathbf{c}) f(x)<0\) over the interval \([0,2 \pi)\) $$f(x)=\tan ^{2} x-3$$