Label the points as follows:
O - the origin (0,0)
A - the source of the light (a,0)
B - the intersection of the first mirror
and the x-axis (b,0)
C - the reflection point at the second mirror
D - the reflection point at the first mirror
ACDA is the path the light beam travels.
Using the sine rule,
|AO| |AC|
----------- = -----------
sin(<ACO) sin(<AOC)
|AC| |AD|
----------- = -----------
sin(<ADC) sin(<ACD)
|AD| |AB|
----------- = -----------
sin(<ABD) sin(<ADB)
If t is the measure of <AOC, then
a |AC|
------------ = --------
sin(135-t) sin(t)
|AC| |AD|
------------- = ------------
sin(360-4t) sin(2t-90)
|AD| b-a
--------- = ------------
sin(45) sin(2t-90)
Therefore,
a sin(t)
---------------
|AC| sin(360-4t) sin(135-t)
------ = ------------- = -----------------
|AD| sin(2t-90) (b-a) sin(45)
---------------
sin(2t-90)
or
a sin(t) sin(2t-90)2 = (b-a) sin(45) sin(135-t) sin(360-4t)
or
a sin(t) cos(2t)2 = (b-a) ½ [cos(t) + sin(t)] [- sin(4t)]
or
a cos(2t) = 2(a-b) cos(t) [cos(t) + sin(t)]
or
a [1 + 2 cos(t)sin(t] = 2b cos(t) [cos(t) + sin(t)]
or
a [cos(t) + sin(t)]2 = 2b cos(t) [cos(t) + sin(t)]
or
a [cos(t) + sin(t)] = 2b cos(t)
or
a sin(t) = (2b - a) cos(t)
Thus,
sin(t) 2b - a
m = tan(t) = -------- = --------
cos(t) a
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