Linear transformation example using trig functions.

Question

In this example, what gets us from:

$$= \begin{bmatrix} r\cos\theta \cos\alpha - r \sin\theta \sin\alpha \\ r \sin\theta \cos\alpha + r\cos\theta \sin\alpha \end{bmatrix}$$

to

$$= \begin{bmatrix} r\cos(\theta + \alpha) \\ r\sin(\theta + \alpha \end{bmatrix} $$

Answer 1

These are the angle sum identities: $$ \sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha\\ \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta $$