Question : Find the values of the parameter a for which the equation $ 3\sin x -4\cos x = a$ has a solution.
Not looking for the answer here but can somebody please recommend me a book or a website to get sufficient knowledge on this topic so that I'm able to answer this question? (I'm familiar with parametric equations involving only unknowns, but this is my first time seeing trigonometric functions in one)
Thanks in advance.
$$3 \sin x-4 \cos x=a \implies (3/5) \sin x-(4/5) \cos x=a/5\implies \sin (x-t)=a/5, \tan t=4/3~~~~(1)$$ This equation will be valid if $-5 \le a\le 5.$ Solutions of (1) are given by $$x= n\pi+(-1)^n \sin^{-1}(a/5)+t, n \in I$$