Use the diagram to prove the double angle
formula, where t=tanq:
tan2q =
2t1-t2
, sin2q =
2t1+t2
, cos2q =
1-t21+t2
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¢=pcos2q+ qsin2q, q¢=psin2q- qcos2q (1)
where
m=tanq. Hence establish another proof
that the matrix
T2 =
æ ç
è
cos2q
sin2q
sin2q
-cos2q
ö ÷
ø
gives a reflection
in the line y=xtanq.
The point P¢¢=(p¢¢,q¢¢) is the image of the
point P¢ after reflection in the line
y=xtanf. Apply the transformation
T2¢ =
æ ç
è
cos2f
sin2f
sin2f
-cos2f
ö ÷
ø
to the point
P¢=(p¢,q¢) to find the coordinates of the point
P¢¢ in terms of p, q, q and f.
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. q = f)?
