I was doing some problems and decided to graph $\sin(x) + \cos(x)$. To my surprise, I received another sine/cosine function, with some phase shift and change in amplitude!! To be precise, I used a graphing calculator to determine that $\sin(x) + \cos(x) = \sqrt2 \sin(x+(\pi/4))$
I understand that $\cos(x)$ is just a phase shift of $\sin(x)$. But now, I'm really curious about this phenomenon. As I understand, the period of 2 sine functions need to be the same to produce another sine function. I wish to know whether there is a general formula for $A \sin(b(x+c)) + A \sin(b(x+d))$, where $A$ and $b$ are positive real numbers, and $c$ and $d$ can be any real number. [Edit: Just to clarify, I don't mean arcsin. I meant A multiplied by sin.]
Essentially, I'm asking for a formula when you add 2 sine functions with the same amplitude and period, but not necessarily the same phase shift. Thanks a ton! And see if you can justify your answers.
