Are there any identities for trigonometric equations of the form:
$$A\sin(x) + B\sin(y) = \cdots$$
$$A\sin(x) + B\cos(y) = \cdots$$
$$A\cos(x) + B\cos(y) = \cdots$$
I can't find any mention of them anywhere, maybe there is a good reason why there aren't identities for these? Thanks!
Best Answer
there are no general formula for these expressions.but may exist when $A$ and $B$ are interrelated .
For example consider triangle $ABC$ where $a,b,\text{ and }c $ are the sides of the triangle and $A,B,\text{ and }C$ are the respective angles opposite to $a,b,\text{ and }c $ then $$c = a\cos B + b\cos A $$ here this is because $a,b ,A\text{ and }B$ are interrelated by laws of triangle.
therefore random values of the angles and the coefficients will not satisfy to form general formula.