Chapter 14

Find the exact value of each expression by using the half-angle formulas. \(\sin 22 \frac{1}{2}\)

Find the exact value of each expression. \(\cos 225^{\circ}\)

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

Find the value of each expression. \(\csc \theta,\) if \(\cos \theta=-\frac{2}{3} ; 180^{\circ}<\theta<270\)

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

Solve each equation for all values of ? if ? is measured in radians. \(\cos ^{2} \theta-\frac{5}{2} \cos \theta-\frac{3}{2}=0\)

Find the exact value of each expression. \(\sin 315^{\circ}\)

Verify that \(\tan \theta \sin \theta \cos \theta \csc ^{2} \theta=1\) is an identity.

Simplify each expression. \(\cos \theta \csc \theta\)

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\sin \left(\theta-\frac{\pi}{4}\right) $$

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

Solve each equation for all values of ? if ? is measured in radians. \(\cos \theta=3 \cos \theta-2\)

Find the exact value of each expression by using the half-angle formulas. \(\sin 345^{\circ}\)

Find the exact value of each expression. \(\sin \left(-15^{\circ}\right)\)

Show that \(1+\cos \theta\) and \(\frac{\sin ^{2} \theta}{1-\cos \theta}\) form an identity.

Simplify each expression. \(\tan \theta \cot \theta\)

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

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

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

For Exercises 20 and \(21,\) use the following information. If an object is propelled from ground level, the maximum height that it reaches is given by \(h=\frac{v^{2} \sin ^{2} \theta}{2 g},\) where \(\theta\) is the angle between the ground and the initial path of the object, \(v\) is the object's initial velocity, and \(g\) is the acceleration due to gravity, 9.8 meters per second squared. Verify the identity \(\frac{v^{2} \sin ^{2} \theta}{2 g}=\frac{v^{2} \tan ^{2} \theta}{2 g \sec ^{2} \theta}\).

Find the exact value of each expression. \(\cos \left(-45^{\circ}\right)\)

Simplify each expression. \(\sin \theta \cot \theta\)

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\frac{1}{4} \tan \left(\theta+22.5^{\circ}\right) $$

Find the amplitude, if it exists, and period of each function. Then graph each function. \(y=\sec 3 \theta\)

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

Find the exact value of each expression by using the half-angle formulas. \(\cos \frac{7 \pi}{12}\)

For Exercises 20 and \(21,\) use the following information. If an object is propelled from ground level, the maximum height that it reaches is given by \(h=\frac{v^{2} \sin ^{2} \theta}{2 g},\) where \(\theta\) is the angle between the ground and the initial path of the object, \(v\) is the object's initial velocity, and \(g\) is the acceleration due to gravity, 9.8 meters per second squared. A model rocket is launched with an initial velocity of 110 meters per second at an angle of \(80^{\circ}\) with the ground. Find the maximum height of the rocket.

Find the exact value of each expression. \(\cos \left(-150^{\circ}\right)\)

Simplify each expression. \(\cos \theta \tan \theta\)

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=3 \sin \left(\theta-75^{\circ}\right) $$

Find the amplitude, if it exists, and period of each function. Then graph each function. \(y=\cot 5 \theta\)

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

Verify that each of the following is an identity. $$ \sin 2 x=2 \cot x \sin ^{2} x $$

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

Simplify each expression. 2\(\left(\csc ^{2} \theta-\cot ^{2} \theta\right)\)

State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. $$ y=\sin \theta-1 $$

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in degrees. \(4 \sin ^{2} \theta-4 \sin \theta+1=0\)

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

On December \(22,\) the maximum amount of light energy that falls on a square foot of ground at a certain location is given by \(E \sin \left(113.5^{\circ}+\phi\right),\) where \(\phi\) is the latitude of the location. Find the amount of light energy, in terms of \(E,\) for each location. Salem, OR (Latitude: \(44.9^{\circ} \mathrm{N}\))

Simplify each expression. 3\(\left(\tan ^{2} \theta-\sec ^{2} \theta\right)\)

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

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

Verify that each of the following is an identity. $$ \sin ^{4} x-\cos ^{4} x=2 \sin ^{2} x-1 $$

On December \(22,\) the maximum amount of light energy that falls on a square foot of ground at a certain location is given by \(E \sin \left(113.5^{\circ}+\phi\right),\) where \(\phi\) is the latitude of the location. Find the amount of light energy, in terms of \(E,\) for each location. Chicago, IL (Latitude: \(41.8^{\circ} \mathrm{N}\))

Simplify each expression. \(\frac{\cos \theta \csc \theta}{\tan \theta}\)

State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. $$ y=\cos \theta-5 $$

MEDICINE For Exercises 24 and \(25,\) use the following information. If the amplitude of the sine function is \(0.25,\) write the equations for tuning forks that resonate with a frequency of \(64,256,\) and 512 Hertz.

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. $$ \sin ^{2} x=\frac{1}{2}(1-\cos 2 x) $$

On December \(22,\) the maximum amount of light energy that falls on a square foot of ground at a certain location is given by \(E \sin \left(113.5^{\circ}+\phi\right),\) where \(\phi\) is the latitude of the location. Find the amount of light energy, in terms of \(E,\) for each location. Charleston, SC (Latitude: \(28.5^{\circ} \mathrm{N} \))