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# Reflect Again

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Age 16 to 18

Challenge Level

Use the diagram to prove the double angle formula, where $t=\tan \theta$: $$\tan2\theta = {2t\over {1-t^2}},\quad \sin2\theta = {2t\over {1+t^2}},\quad \cos2\theta = {{1-t^2}\over {1+t^2}}$$

The point $P'=(p',q')$ is the image of the point $P=(p,q)$ after reflection in the line $y=mx$. To find $(p',q')$ use the fact that the midpoint of $PP'$ is on the line $y=mx$ and the line segment $PP'$ is perpendicular to the line $y=mx$ and show that $$p'=p\cos2\theta + q\sin2\theta,\ q'=p\sin2\theta - q\cos2\theta\quad (1)$$ where $m=\tan\theta$. Hence establish another proof that the
matrix

$$T_2= \left( \begin{array}{cc} \cos 2\theta &\sin2\theta \\ \sin2\theta &-\cos2\theta \end{array} \right) $$

gives a reflection in the line $y=x\tan\theta$. The point $P''=(p'',q'')$ is the image of the point $P'$ after reflection in the line $y=x\tan\phi$. Apply the transformation $$T_2' = \left(\begin{array}{cc} \cos 2\phi &\sin2\phi \\ \sin2\phi & -\cos2\phi\end{array}\right)$$ to the point $P'=(p',q')$ to find the coordinates of the point $P''$ in terms of $p, q, \theta$ and $\phi$. Hence
show that the combination of two reflections in distinct intersecting lines is a rotation about the point of intersection by twice the angle between the two mirror lines. What is the effect of the two reflections if the lines coincide (i.e. $\theta=\phi$)?

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Investigate the transformations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0, -1 and +1.