Chapter 7: Analytic Trigonometry

(a) Show that$\tan \left({\tan }^{-1}1+{\tan }^{-1}2+{\tan }^{-1}3\right)=0$

(b) Conclude from part (a) that

$\tan \left({\tan }^{-1}1+{\tan }^{-1}2+{\tan }^{-1}3\right)=0$

A searchlight at the grand opening of a new car dealership casts a spot of light on a wall located 75 meters from the searchlight. The acceleration r of the spot of light is found to be $r=1200sec\theta (2se{c}^2\theta -1)$. Show that this is equivalent to$r=1200\left(\frac{1+{\sin }^2\theta }{{\cos }^3\theta }\right)$

If $z=\tan \frac{\alpha }{2}$, show that $\sin \alpha =\frac{2z}{1+z^2}$.

Holding Pattern An airplane is asked to stay within a holding pattern near Chicago’s O’Hare International Airport. The function $d\left(x\right)=70\sin (0.65x)+150$ represents the distance d, in miles, of the airplane from the airport at time x, in minutes.

(a) When the plane enters the holding pattern, x = 0, how far is it from O’Hare?

(b) During the first 20 minutes after the plane enters the holding pattern, at what time x is the plane exactly 100miles from the airport?

(c) During the first 20 minutes after the plane enters the holding pattern, at what time x is the plane more than 100 miles from the airport?

(d) While the plane is in the holding pattern, will it ever be within 70 miles of the airport? Why?

Projectile Motion A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range R of the ball as a function of the angle $\theta$to the horizontal is given by $R\left(\theta \right)=672\sin \left(2\theta \right),$ where R is measured in feet.

(a) At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 450 feet (150 yards)?

(b) At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 540 feet (180 yards)?

(c) At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel at least 480 feet (160 yards)?

(d) Can the golfer hit the ball 720 feet (240 yards)?

In an alternating current (ac) circuit, the

instantaneous power p at time t is given by

$p\left(t\right)=v_m{i}_m\cos \varphi {\sin }^2\left(\omega t\right)-v_m{i}_m\sin \varphi \sin \left(\omega t\right)\cos \left(\omega t\right)$

Show that this is equivalent to

$P\left(t\right)=v_m{i}_m\sin \left(\omega t\right)\sin (\omega t-\varphi )$

If $z=\tan \frac{\alpha }{2}$, show that localid="1646416257885" $\cos \alpha =\frac{1-z^2}{1+z^2}$.

Optical methods of measurement often rely on the interference of two light waves. If two light waves, identical except for a phase lag, are mixed together, the resulting intensity, or irradiance, is given by $I_1=4A^2\frac{(csc\theta -1)(sec\theta +\tan \theta )}{csc\theta sec\theta }$

Show that this is equivalent to$I_1={(2A\cos \theta )}^2$

Write a few paragraphs outlining your strategy for establishing identities.

Graph $f\left(x\right)={\sin }^{2}x=\frac{1-\cos \left(2x\right)}{2}$for $0\le x\le 2\pi$by hand using transformations.