From Förberedande kurs i matematik 1
If we want to solve the equation
cos 3x=sin 4x, we need an additional result which tells us for which values of
u
and
v
the equality
cos u=sin v
holds, but to get that we have to start with the equality
cosu=cosv.
So, we start by looking at the equality
cosu=cosv
We know that for fixed
u
there are two angles
v=u
and
v=−u
in the unit circle which have the cosine value
cosu, i.e. their
x
-coordinate is equal to
cosu.
Imagine now that the whole unit circle is rotated anti-clockwise an angle
2. The line
x=cosu
will become the line
y=cosu
and the angles
u
and
−u
are rotated to
u+2
and
−u+2, respectively.
The angles
u+2
and
−u+2
therefore have their
y
-coordinate, and hence sine value, equal to
cosu. In other words, the equality
cos u=sin v
holds for fixed
u
in the unit circle when
v=
u+2, and more generally when
v=u+2+2n
(
n
an arbitrary integer).
For our equation
cos 3x=sin 4x, this result means that
x
must satisfy
4x=3x+2+2n
This means that the solutions to the equation are
x=
2+2nx=14+72n
(
\displaystyle n
an arbitrary integer)
