Shoot a basketball with a minimum speed at some angle

[McFluffy]

Homework Statement

You should shoot a basketball at the angle $\theta$ requiring minimum speed. Avoid line drives and rainbows. Shooting from (0, 0) with the basket at (a, b), minimize $f(\theta)= 1/(a \sin (\theta) \cos (\theta) -b \cos^2 (\theta))$.
(a) If b =0 you are level with the basket. Show that $\theta=45\deg$ is best (Jabbar sky hook).
(b) Reduce $df\over d\theta$ = 0 to $\tan (2\theta) = -a/b$. Solve when a =b.
(c) Estimate the best angle for a free throw

Homework Equations

I have no idea on how this derived:$f(\theta)= 1/(a \sin (\theta) \cos (\theta) -b \cos^2 (\theta))$, but based on the question, it seems that this is the initial speed $f$, as a function of angle, $\theta$? How does one derive this?

The Attempt at a Solution

Coming from a person that only plays basketball when friends invited to play, I don't understand the terminologies that are being used such as, "line drives", "rainbows", and "Jabbar sky hook". So the first thing I did was to google these phrases:
Line drives = a flat shot with no arc, which is a straight line from (0,0) to (a,b)
Rainbows = A type of shot in basketball which has a higher than normal arc as it moves towards the basket, so it's a parabola.
Jabbar sky hook = When you're level at the basket, you throw the ball, this video explains it:

Now that we know the terminologies(I think), we can proceed. So a straight line or a parabolic path is a no. The question first asked me to minimize $f(\theta)$, setting $df \over d\theta$ = 0 then finding what $\theta$ to plug into $f(\theta)$ is fairly straightforward but I want to know how was $f(\theta)$ derived? I tried sketching what it would look like in the xy plane and find some clues on how to derive $f(\theta)$ but I can't find it.

Other than that,the part where I'm stuck is the formulation part. The calculus part, I can do those. So what now?

 

[kuruman]

How does one derive this?

Eliminate the time from the two projectile equations and get $y(x)$. Be sure to write the trig functions in the equation in terms of Tan(θ), then solve for v₀². You will get $f(\theta)$ to within a constant which is irrelevant because you are minimizing.

On edit: Actually you don't have to express the trig functions in terms of the tangent. That becomes necessary if you want to find the angle of projection given a, b and the initial speed.