From Förberedande kurs i matematik 1
If we use the Pythagorean identity and write
sin2x
as
1−cos2x, the whole equation written in terms of
cosx
becomes
2
1−cos2x
−3cosx=0
or, in rearranged form,
2cos2x+3cosx−2=0
With the equation expressed entirely in terms of
cosx, we can introduce a new unknown variable
t=cosx
and solve the equation with respect to
t. Expressed in terms of
t, the equation is
2t2+3t−2=0
and this second-degree equation has the solutions
t=21
and
t=−2
.
In terms of
x, this means that either
cosx=21
or
cos x=−2. The first case occurs when
x=
3+2n
(
n
an arbitrary integer),
whilst the equation
cos x=−2
has no solutions at all (the values of cosine lie between
−1
and
1
).
The answer is that the equation has the solutions
x=3+2n
(
n
an arbitrary integer).
