91Ó°ÊÓ

If \(x=2 \tan \theta,\) express \(\sin (2 \theta)\) as a function of \(x\)

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 \(R(\theta)=672 \sin (2 \theta),\) 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)?

The horizontal distance that a projectile will travel in the air (ignoring air resistance) is given by the equation $$ R(\theta)=\frac{v_{0}^{2} \sin (2 \theta)}{g} $$ where \(v_{0}\) is the initial velocity of the projectile, \(\theta\) is the angle of elevation, and \(g\) is acceleration due to gravity (9.8 meters per second squared). (a) If you can throw a baseball with an initial speed of 34.8 meters per second, at what angle of elevation \(\theta\) should you direct the throw so that the ball travels a distance of 107 meters before striking the ground? (b) Determine the maximum distance that you can throw the ball. (c) Graph \(R=R(\theta),\) with \(v_{0}=34.8\) meters per second. (d) Verify the results obtained in parts (a) and (b) using a graphing utility.

If an angle \(\theta\) lies in quadrant III and \(\cot \theta=\frac{8}{5},\) find \(\sec \theta\)