What does adding $\sin\theta \cos\theta$ make my graph a linear relationship?

Question

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.

Answer 1

As θ goes from 0 to 90
Sinθ goes from 0 to 1 (increasing)
Cosθ goes from 1 to 0 (decreasing)
SinθCosθ goes from 0 to 0 & with maximum value of 0.5 at 45.

So it increases from 0 to 0.5 and decreases again to 0, which is similar to the range. That's why the Range vs SinθCosθ graph is linear.

It's like this: in Range vs SinθCosθ graph, Imagine a ball at (0,0) when θ = 0, as θ starts increasing the value of both Range & SinθCosθ will increase till θ = 45, i.e. the ball is moving up on the straight line. At this, the value of SinθCosθ = 0.5 & Range = 10 ( Maximum values)

As θ is moving from 45 to 90, the value of both Range & SinθCosθ decreases, i.e. the ball is sliding down on the straight line.

Answer 2

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 $\pi/2$.