The same angle allows the largest margin of error (_Sports Science_ by…

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 $\theta =45^{\circ }$ is best (Jabbar sky hook).

(b) Reduce $df/d\theta =0$ to $\tan 2\theta =-a/b$. Solve when $a=b$.

(c) Estimate the best angle for a free throw.

The same angle allows the largest margin of error (_Sports Science_ by Peter Brancazio). Section 12.2 gives the flight path.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.2 Maximum and Minimum Problems, #27