From Förberedande kurs i matematik 1
If we consider for a moment the equality
sinu=sinv(
)
where
u
has a fixed value, there are usually two angles
v
in the unit circle which ensure that the equality holds:
v=u
and
v=
−u
(The only exception is when
u=
2
or
u=32 , in which case
u
and
−u
correspond to the same direction and there is only one angle
v
which satisfies the equality.)
We obtain all the angles
v
which satisfy (*) by adding multiples of
2,
v=u+2n
and
v=−u+2n
where
n
is an arbitrary integer.
If we now go back to our equation
sin3x=sinx
the reasoning above shows that the equation is only satisfied when
3x=x+2n
or
3x=−x+2n
If we make
x
the subject of each equation, we obtain the full solution to the equation:
x=0+nx=
4+21n
