What does adding $\sin\theta \cos\theta$ make my graph a linear relationship? What is the point of adding sin n cos of theta when graphing range?
e.g. I see on hyperphysics a graph of range vs sin n cos of theta and it makes the experimental data embody a linear relationship. In contrast to range vs angle, you get a hyperbolic sort of shape.



Projectile equation:
$$y=x\tan\theta-\frac{gx^2}{2v^2}\sec^2\theta$$
When $y=0$ (i.e. at launch and at landing), 
$$\begin{align}
x\left(\frac{gx}{2v^2}\sec^2\theta-\tan\theta\right)&=0\\
x&=0,\frac{2v^2}g \frac{tan\theta}{\sec^2\theta}\end{align}$$
Hence range
$$R=\frac{v^2}g \frac{2\tan\theta}{\sec^2\theta}=\frac{v^2}g(2\sin\theta\cos\theta)=\frac{v^2}g\sin2\theta
$$
Hence $R$ varies linearly with $\sin\theta\cos\theta$. 
Also, $\sin2\theta$ (and hence $R$) varies sinusoidally from $0$ to $1$ to $0$, as $\theta$ varies from $0$ to $\pi/4$ to $\pi/2$. 
