# Thread: matrix transformation for trig identities

1. ## matrix transformation for trig identities

Hi folks,

Given that the following matrix creates an anticlockwise rotation of the x-y plane about the origin through an angle $\displaystyle \theta$:

$\displaystyle \left( \begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right)$

use this fact to obtain the standard trig identities:

$\displaystyle \sin(\theta + \alpha) = \sin\theta\cos\alpha + cos\theta\sin\alpha$ and

$\displaystyle \cos(\theta + \alpha) = \cos\theta\cos\alpha - \sin\theta\sin\alpha$

I can obtain the identities using geometry and I can see how the matrix transformation creates an anticlockwise rotation, but I can't see how to use the matrix trransformation to generate the identities. Can anyone help?

2. Calculate

$\displaystyle \left( \begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right) \left( \begin{array}{cc} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{array} \right)$

and note that this is equivalent to rotating through by $\displaystyle \theta + \alpha$, which is the matrix

$\displaystyle \left( \begin{array}{cc} \cos(\theta + \alpha) & -\sin(\theta + \alpha) \\ \sin(\theta + \alpha) & \cos(\theta + \alpha) \end{array} \right)$

3. Ahhhhh! Now I see. Thanks very much Icemanfan.