Processing Math: Done

From Förberedande kurs i matematik 1

If we use the Pythagorean identity and write sin2x as 1−cos2x, the whole equation can be written in terms of cosx,

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 quadratic equation has the solutions t=21 and t=−2.

In terms of x, this means that either cosx=21 or cosx=−2. The first case occurs when

x=3+2n(n is an arbitrary integer),

whilst the equation cosx=−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

where n is an arbitrary integer.