91Ó°ÊÓ

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin 2 x=2 \cos ^{2} x$$

Use a graphing calculator to make a conjecture about whether each equation is an identity. $$\cos 2 x=1-2 \sin ^{2} x$$

Verify that each equation is an identity. $$\sec ^{2} \frac{x}{2}=\frac{2}{1+\cos x}$$

Use a calculator to find each value. Give answers as real numbers. $$\cos \left(\tan ^{-1} 0.5\right)$$

Verify that each equation is an identity. $$\frac{2}{1+\cos x}-\tan ^{2} \frac{x}{2}=1$$

The model $$ 0.342 D \cos \theta+h \cos ^{2} \theta=\frac{16 D^{2}}{V_{0}^{2}} $$ is used to reconstruct accidents in which a vehicle vaults into the air after hitting an obstruction. \(V_{0}\) is velocity in feet per second of the vehicle when it hits the obstruction, \(D\) is distance (in feet) from the obstruction to the landing point, and \(h\) is the difference in height (in feet) between landing point and takeoff point. Angle \(\theta\) is the takeoff angle, the angle between the horizontal and the path of the vehicle. Find \(\theta\) to the nearest degree if \(V_{0}=60, D=80,\) and \(h=2\)

The following function approximates the average monthly temperature \(y\) (in \(^{\circ} \mathrm{F}\) ) in Phoenix, Arizona. Here \(x\) represents the month, where \(x=1\) corresponds to January, \(x=2\) corresponds to February, and so on. $$ f(x)=19.5 \cos \left[\frac{\pi}{6}(x-7)\right]+70.5 $$ When is the average monthly temperature (a) \(70.5^{\circ} \mathrm{F}\) (b) \(55^{\circ} \mathrm{F} ?\)

The study of alternating electric current requires the solutions of equations of the form $$ i=I_{\max } \sin 2 \pi f t $$ for time t in seconds, where is instantaneous current in amperes. \(I_{\max }\) is maximum cur. rent in amperes, and F is the number of cycles per second.Find the least positive value of \(t,\) given the following data. $$i=\frac{1}{2} I_{\max }, f=60$$