Chapter 7, Problem 12E

To plot $y=\sin \left(x\right),$ we must define vectors of x and y values and then use the plot command. This can be done as follows:
$\begin{array}{l}x=0:0.1:6;y=\sin \left(x\right);\\ \text{plot}\left(x,y\right)\end{array}$

1. Let us define a rotation matrix and use it to rotate the graph of $y=\sin \left(x\right).$ Set
   $\begin{array}{l}t=\text{pi/4;}c=\cos \left(t\right);s=\sin \left(t\right);\\ R=\left[c,-s;s,c\right]\end{array}$
   To find the rotated coordinates, set
   $Z=R*\left[x;y\right];x1=Z(1,:);y1=Z(2,:);$
   The vectors x1 and y1 contain the coordinates for the rotated curve. Set
   $w=\left[0,5\right];\text{axis}\text{square}$
   And plot x1 and y1, using the MATLAB command
   $\text{plot}\left(x1,y1,w,w\right)$
   By what angles has the graph been rotated and in what direction?
2. Keep all your variables from part(a) and set
   $G=\left[c,s;s,c\right]$
   The matrix G represents a Given reflection. To determine the reflected coordinates, set
   $\begin{array}{l}z=G*\left[x;y\right];\\ x2=Z\left(1,:\right);y2=Z\left(2,:\right);\end{array}$
   Plot the reflected curve, using the MATLAB command
   $\text{plot}\left(x2,y2,w,w\right)$
   The curve $y=\sin \left(x\right)$ has been reflected about a line through the origin making an angle of $\pi /8$ with the x-axis. To see this, set
   $\begin{array}{l}w1=\left[0,6.3*\cos \left(t/2\right)\right];\\ z1=\left[0,6.3*\sin \left(t/2\right)\right];\end{array}$
   and plot the new axis and both curves with the MATLAB command
   $\text{plot}\left(x,y,x2,y2,w1,z1\right)$
3. Use the rotation matrix R from part (a) to rotate the curve $y=-\sin \left(x\right).$ Plot the rotated curve. How does the graph compare to that of the curve from part (b)? Explain.