Aus Online Mathematik Brückenkurs 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
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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
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holds for fixed u in the unit circle when v=
u+2, and more generally when
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For our equation cos3x=sin4x, this result means that x must satisfy
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This means that the solutions to the equation are
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where n is an arbitrary integer.
