Chapter 5, Problem 5.43P

Find the distribution of $R=A\sin \theta$, where A is a fixed constant and $\theta$ is uniformly distributed on $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Such a random variable R arises in the theory of ballistics. If a projectile is fired from the origin at an angle $\alpha$ from the earth with a speed v. then the point R at which it returns to the earth can be expressed as $R=\left(\frac{v^2}{g}\right)\sin 2\alpha$, where g is the gravitational constant, equal to 980 centimeters per second squared.