How is the formula to determine the angle to launch a projectile to hit a coordinate derived?

Question

Wikipedia lists this formula as

Angle θ required to hit coordinate (x, y)

It is used to determine the angle at which to launch a projectile in order to hit a coordinate. How is it derived?

user256872 · Accepted Answer

Start off with the following system of equations describing the $x$ and $y$ motion launched from the origin:
$$
left{ 
x=v_0tcostheta atop
y=-dfrac{1}{2} gt^2 + v_0 tsintheta
right.
$$
Solving for $t$ in the $x$ equation and substituting into the $y$ equation,
$$implies y = -dfrac{g}{2v_0^2 cos^2theta} x^2 + xtantheta$$
From there, I rearrange to get the following
$$
implies 0 = -dfrac{gx^2}{2v_0^2} sec^2theta+ xtantheta - y
$$
Recall the $tan^2thetaequiv sec^2theta-1$ identity, and use it. Answer should be straightforward from there.

G. Smith · Answer

Write $x(t)$ and $y(t)$ for the projectile. Eliminate $t$ to get an equation relating $x$, $y$, $v$ and $theta$. Solve it for $theta$ in terms of $x$, $y$, and $v$. You will need to use a trigonometric identity to express the equation only in terms of $tantheta$, and then solve this quadratic equation.

Answered by G. Smith on January 22, 2021

How is the formula to determine the angle to launch a projectile to hit a coordinate derived?

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