I am trying to solve a physics problem to do with finding the ideal angle for the maximum range, $x$, of a projectile, with air resistance taken into consideration ($\therefore \theta \neq 45^{\circ}$).
Here is the final equation I have arrived at:
$$x =\frac{u^2\sin(2\theta)}{g}-\frac{2Fu^2\sin^2(\theta)}{mg^2}$$
How would I find the angle $\theta$ that results in the greatest possible value of $x$, assuming that the values of $u$, $g$, $F$, and $m$ are known constants?
Let $\alpha = \frac{u^2}{g}$ and $\beta = \frac{2Fu^2}{mg^2}$ \begin{equation} \frac{d x}{d \theta} = 2\alpha\cos(2\theta) - 2 \beta \sin \theta \cos \theta = 0 \end{equation} But $\sin 2\theta =2 \sin \theta \cos \theta $, hence arranging we get $$\tan 2 \theta = \frac{2\alpha}{\beta}$$ Can you continue?