# Math Help - Matrix

1. ## Matrix

HI, how can solved this question? What sort of eqn should i form? Right angle triangle eqn? Thanks

2. I copied your question and the graph into the following figure, please see the attachment. consider the given matrix system, you are asking to prove that: $x_2=x_1\cos\theta + y_1\sin\theta$ and $y_2 = -x_1\sin\theta + y_1\cos\theta$.

According to your description and the attached figure, we know that $x_1=OT,\;y_1=PT$ and $x_2=PR=OQ,\;y_2=PQ$. So what we need to show is $OQ=(OT)\cos\theta + (PT)\sin\theta$ and $PQ = -(OT)\sin\theta + (PT)\cos\theta$. Next I am going to show these two identities actually hold, all we need is just a few manipulations of the line segments and trig. definitions. Please consider the two right triangles $\triangle OST$ and $\triangle PSQ$.

First, we have
$PQ=(PS)\cos\theta=(PT-ST)\cos\theta=\left[PT-(OS)\sin\theta\right]\cos\theta$
$=\left[PT-\left(\frac{OT}{\cos\theta}\right)\sin\theta\right]\cos\theta= -(OT)\sin\theta + (PT)\cos\theta$

Then, consider
$OQ=OS+SQ=\frac{OT}{\cos\theta}+(PS)\sin\theta=\fra c{OT}{\cos\theta}+(PT-ST)\sin\theta$
$=\frac{OT}{\cos\theta}+(PT-(OT)\tan\theta)\sin\theta=\frac{OT}{\cos\theta}+(P T)\sin\theta-(OT)\frac{\sin^2\theta}{\cos\theta}$
$=(OT)\cos\theta + (PT)\sin\theta$

This completes the proof.

Roy

3. With reference to the same diagram,

If matrix A = (cos theta sin theta)
( -sin theta cos theta)

1)what form of coordinate axis rotation does A^3 represent?

2)what form of coordinate axis rotation does A^(-1) represent?