From Förberedande kurs i matematik 1
Let's first investigate when the equality
tanu=tanv
is satisfied. Because
u
can be interpreted as the slope (gradient) of the line which makes an angle
u
with the positive
x
-axis, we see that for a fixed value of tan u, there are two angles
v
in the unit circle with this slope:
v=u
and
v=u+
slope
= tan u
slope
= tan u
The angle
v
has the same slope after every half turn, so if we add multiples of
to
u, we will obtain all the angles
v
which satisfy the equality
v=u+n
where
n
is an arbitrary integer.
If we apply this result to the equation
tanx=tan4x
we see that the solutions are given by
4x=x+n
(
n
an arbitrary integer),
and solving for
x
gives
x=31n
(
n
an arbitrary integer).
