matrix transformation for trig identities

**s_ingram** · February 16th 2010, 10:47 AM

Hi folks,

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

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

use this fact to obtain the standard trig identities:

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

$\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?

**icemanfan** · February 16th 2010, 10:56 AM

Calculate

$ \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 $\theta + \alpha$ , which is the matrix

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

**s_ingram** · February 16th 2010, 10:59 AM

Ahhhhh! Now I see. Thanks very much Icemanfan.

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