Chapter 10

Use identities to write each expression as a function with \(x\) as the only argument. $$\sin \left(x-90^{\circ}\right)$$

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

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

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. $$2-4 \sin ^{2} 15^{\circ}$$

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

Use identities to write each expression as a function with \(x\) as the only argument. $$\sin \left(x+90^{\circ}\right)$$

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. $$2 \sin \theta=2 \cos 2 \theta$$

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

Graph each function and use the graph to make a conjecture about what might be an identity. Then verify your conjecture. $$f(x)=\cos ^{4} x-\sin ^{4} x$$

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

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

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. $$\cos \theta-1=\cos 2 \theta$$

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

Solve \((\mathbf{a}) f(x)=0,(\mathbf{b}) f(x)>0,\) and \((\mathbf{c}) f(x)<0\) over the interval \([0,2 \pi)\) $$f(x)=2 \sin ^{2} x+3 \sin x+1$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan \left(360^{\circ}-x\right)$$

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. $$1-\sin x=\cos 2 x$$

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

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

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.) $$\tan \theta \cos \theta$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(\frac{\pi}{2}-x\right)$$

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. $$\sin 2 x=2 \cos ^{2} x$$

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

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

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.) $$\cot \alpha \sin ^{2} \alpha \csc \alpha$$

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

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. $$3 \csc ^{2} \frac{x}{2}=2 \sec x$$

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

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

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.) $$\frac{\sin \beta \tan \beta}{\cos \beta}$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(\frac{3 \pi}{2}+x\right)$$

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. $$\cos x=\sin ^{2} \frac{x}{2}$$

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

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

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.) $$\frac{\csc \theta \sec \theta}{\cot \theta}$$

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

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. $$2-\sin 2 \theta=4 \sin 2 \theta$$

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

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\sin \frac{\pi}{12}$$

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

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.) $$\sec ^{2} x-1$$

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

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. $$4 \cos 2 \theta=8 \sin \theta \cos \theta$$

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

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\cos \frac{\pi}{8}$$

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

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.) $$\csc ^{2} t-1$$

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan (2 \pi-x)$$

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. $$2 \cos ^{2} 2 \theta=1-\cos 2 \theta$$

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\arcsin \left(-\frac{\sqrt{3}}{2}\right)$$

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\tan \left(-\frac{\pi}{8}\right)$$