Processing Math: Done

From Förberedande kurs i matematik 1

If we want to solve the equation cos3x=sin4x, we need an additional result which tells us for which values of u and v the equality cosu=sinv 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

cosu=sinv

holds for fixed u in the unit circle when v=u+2, and more generally when

v=u+2+2n(n is an arbitrary integer).

For our equation cos3x=sin4x, this result means that x must satisfy

4x=3x+2+2n.

This means that the solutions to the equation are

xx=2+2n=14+72n

where n is an arbitrary integer.