Use the diagram to prove the double angle
formula, where t=tanθ:
tan2θ =
2t
1−t2
, sin2θ =
2t
1+t2
, cos2θ =
1−t2
1+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 ' =pcos2θ+ qsin2θ, q ' =psin2θ− qcos2θ (1)
where
m=tanθ. Hence establish another proof
that the matrix
T2=
cos2θ
sin2θ
sin2θ
−cos2θ
gives a reflection
in the line y=xtanθ.
The point P ' ' =(p ' ' ,q ' ' ) is the image of the
point P ' after reflection in the line
y=xtanφ. Apply the transformation
T ' 2=
cos2φ
sin2φ
sin2φ
−cos2φ
to the point
P ' =(p ' ,q ' ) to find the coordinates of the point
P ' ' in terms of p, q, θ and φ.
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. θ = φ)?
