This is what I meant to write and is what I have been trying to find examples of but I can only seem to find definitions of inverse functions or trig substitution used to integrate and not for solving trig equations.

The only thing I seem to spot is that the squares of the coefficients on the LHS add up to a clean 169 so if you divided both sides by 13, you would you have the squares of the coefficients = 1.

How would you apply substitution or the properties of inverse functions to the eqation:

$\displaystyle 5cos(x) + 12sin(x) = 3$

Never thought I'de like trig, but this has got me interested again