# matrix transformation for trig identities

• Feb 16th 2010, 11:47 AM
s_ingram
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 $\theta$:

$\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?
• Feb 16th 2010, 11:56 AM
icemanfan
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)$
• Feb 16th 2010, 11:59 AM
s_ingram
Ahhhhh! Now I see. Thanks very much Icemanfan.