The Exercise

A search runs over $N$ items, $M$ of which are solutions. Let $|\alpha\rangle$ be the normalized uniform superposition of the $N-M$ items that are not solutions, and $|\beta\rangle$ that of the $M$ items that are. The uniform superposition over all $N$ items is then $$|\psi\rangle = \sqrt{\frac{N-M}{N}}\,|\alpha\rangle + \sqrt{\frac{M}{N}}\,|\beta\rangle .$$(1) Both amplitudes are nonnegative and their squares add to one, so they are the cosine and the sine of a half angle: $$\cos\frac{\theta}{2} = \sqrt{\frac{N-M}{N}}, \qquad \sin\frac{\theta}{2} = \sqrt{\frac{M}{N}} .$$(2) Each Grover iteration turns the state through the angle $\theta$ in the plane of $|\alpha\rangle$ and $|\beta\rangle$, and the exercise asks for $\sin\theta$ in terms of $N$ and $M$.

The Claim

What Is Checked

The sine, cosine and square root are the real ones.