While trying to find if a given function is periodic and if so, find its period (later, I proved that it does not have a period - through a different method) I encountered this equation which I was unable to solve (i.e. find the values of $x$ (complex and real) that satisfy it) - $$ \sin(1)=\sin(1+x)-\sin(\pi x) $$
My Approach: Recently, while going through the Wikipedia article on "Sine", I came across this definition of $\sin(x)$ and $\cos(x)$ (for all complex $x$)- $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$ $$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$$
So, I tried using putting these values into the above equation and trying to solve it, but nothing came of it.
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Now, I learnt - through other questions on this site - that such an equation does not necessarily have an algebraic solution; but I was unable to find a question (or any such material elsewhere on the net) that gives a method to solve such equations - $$ \displaystyle\sum_{i=0}^n {a_i(\sin(k_ix))^{p_i}}=0$$
Where $a_i, k_i, p_i$ take all real values.
Using the above mentioned substitutions, the above should become equivalent to solving -
$$\displaystyle\sum_{i=0}^n{m_ie^{t_ix}}=0$$ Where $m_i, t_i$ can take all complex values.
So, my question is - How would one go about solving equations of the above kind in general (or - is there no general method?).
Also, what good book (meaning easy to understand, widely used and having an in-depth coverage of the topic) at an undergraduate level (I'm in the last year of high school) would cover the above kind of equations?