How to calculate the coefficients of a qubit from the angles of its Bloch representation?

Question

A quantum bit $|\psi\rangle=a|0\rangle+b|1\rangle$ is represented on the Bloch sphere as a point on the spherical surface with $\theta = 40^°$ and $\phi = 245^°$. Calculate the (complex) coefficients $a$ and $b$.

Here is theory from wikipedia, that I also looked up

It would be nice to have a step-by-step guide based on a concrete example for reverse operation like it was made here: How can I find the $\theta$ and $\phi$ values of a qubit on the Bloch sphere?

Answer 1

The answer is in the first link:

$$ |ψ⟩=\cos\left(\frac{θ}{2}\right)|0⟩+e^{iϕ} \sin\left(\frac{θ}{2}\right)|1⟩ $$

Identify common coefficients: $$ a = \cos\left(\frac{θ}{2}\right) $$ $$ b = e^{iϕ} \sin\left(\frac{θ}{2}\right) $$

Then simply plug in your angles $θ$ and $ϕ$. Be sure your calculator knows whether you are inputting your angles as radians or degrees (eg. $40°=\frac{2π}{9}$ rad)!