Chapter 14

Find all solutions of each equation for the given interval. \(4 \cos ^{2} \theta=1 ; 0^{\circ} \leq \theta<360^{\circ}\)

Find the exact values of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. $$ \cos \theta=\frac{3}{5} ; 0^{\circ}<\theta<90^{\circ} $$

Verify that each of the following is an identity. $$ \tan \theta(\cot \theta+\tan \theta)=\sec ^{2} \theta $$

Find the value of each expression. $$ \tan \theta, \text { if } \sin \theta=\frac{1}{2} ; 90^{\circ} \leq \theta<180^{\circ} $$

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\sin \left(\theta-\frac{\pi}{2}\right) $$

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=\frac{1}{2} \sin \theta $$

Find all solutions of each equation for the given interval. \(2 \sin ^{2} \theta-1=0 ; 90^{\circ}<\theta<270^{\circ}\)

Find the exact values of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. $$ \cos \theta=-\frac{2}{3} ; 180^{\circ}<\theta<270^{\circ} $$

Find the exact value of each expression. \(\sin 165^{\circ}\)

Find the value of each expression. $$ \csc \theta, \text { if } \cos \theta=-\frac{3}{5} ; 180^{\circ} \leq \theta<270^{\circ} $$

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\tan \left(\theta+60^{\circ}\right) $$

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=2 \sin \theta $$

Find all solutions of each equation for the given interval. \(\sin 2 \theta=\cos \theta ; 0 \leq \theta<2 \pi\)

Find the exact values of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. $$ \sin \theta=\frac{1}{2} ; 0^{\circ}<\theta<90^{\circ} $$

Find the exact value of each expression. \(\cos 255^{\circ}\)

Verify that each of the following is an identity. $$ \frac{\cos ^{2} \theta}{1-\sin \theta}=1+\sin \theta $$

Find the value of each expression. $$ \cos \theta, \text { if } \sin \theta=\frac{4}{5} ; 0^{\circ} \leq \theta<90^{\circ} $$

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\cos \left(\theta-45^{\circ}\right) $$

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=\frac{2}{3} \cos \theta $$

Find all solutions of each equation for the given interval. \(3 \sin ^{2} \theta-\cos ^{2} \theta=0 ; 0 \leq \theta<\frac{\pi}{2}\)

Find the exact values of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. $$ \sin \theta=-\frac{3}{4} ; 270^{\circ}<\theta<360^{\circ} $$

Find the exact value of each expression. \(\cos \left(-30^{\circ}\right)\)

Verify that each of the following is an identity. $$ \frac{1+\tan ^{2} \theta}{\csc ^{2} \theta}=\tan ^{2} \theta $$

Find the value of each expression. $$ \sec \theta, \text { if } \tan \theta=-1 ; 270^{\circ}<\theta<360^{\circ} $$

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\sec \left(\theta+\frac{\pi}{3}\right) $$

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=\frac{1}{4} \tan \theta $$

Solve each equation for all values of ? if ? is measured in radians. \(\cos 2 \theta=\cos \theta\)

Find the exact value of each expression by using the half-angle formulas. \(\sin 195^{\circ}\)

Find the exact value of each expression. \(\sin \left(-240^{\circ}\right)\)

Verify that each of the following is an identity. $$ \frac{\sin \theta}{\sec \theta}=\frac{1}{\tan \theta+\cot \theta} $$

Simplify each expression. $$ \csc \theta \cos \theta \tan \theta $$

State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. $$ y=\cos \theta+\frac{1}{4} $$

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=\csc 2 \theta $$

Solve each equation for all values of ? if ? is measured in radians. \(\sin \theta+\sin \theta \cos \theta=0\)

Find the exact value of each expression by using the half-angle formulas. \(\cos \frac{19 \pi}{12}\)

Find the exact value of each expression. \(\cos \left(-120^{\circ}\right)\)

Verify that each of the following is an identity. $$ \frac{\sec \theta+1}{\tan \theta}=\frac{\tan \theta}{\sec \theta-1} $$

Simplify each expression. $$ \sec ^{2} \theta-1 $$

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=4 \sin 2 \theta $$

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in degrees. \(\sin \theta=1+\cos \theta\)

AVIATION When a jet travels at speeds greater than the speed of sound, a sonic boom is created by the sound waves foom ing a cone behind the jet. If \(\theta\) is the measure of the angle at the vertex of the cone, then the Mach number \(M\) can be determined using the formula \(\sin \frac{\theta}{2}=\frac{1}{M}\) . Find the Mach number of a jet if a sonic boom is created by a cone with a vertex angle of \(75^{\circ} .\)

Simplify each expression. $$ \frac{\tan \theta}{\sin \theta} $$

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in degrees. \(2 \cos ^{2} \theta+2=5 \cos \theta\)

Verify that each of the following is an identity. $$ \cot x=\frac{\sin 2 x}{1-\cos 2 x} $$

Simplify each expression. $$ \sin \theta\left(1+\cot ^{2} \theta\right) $$

State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. $$ y=\sin \theta+0.25 $$

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=\frac{1}{2} \sec 3 \theta $$

Solve each equation for all values of \(\theta\). \(2 \sin ^{2} \theta-3 \sin \theta-2=0\)

Verify that each of the following is an identity. \(\sin \left(\theta+\frac{\pi}{2}\right)=\cos \theta\)

Verify that each of the following is an identity. $$ \cot \theta\left(\cot \theta+\tan ^{2} \theta\right)=\csc ^{2} \theta $$