Is there a nice way to simplify $2\sin(x + \pi/5) + 3\sin(x + \pi/7) $?

Question

$$2\sin(x + \pi/5) + 3\sin(x + \pi/7) $$

I'm searching for all the ways to simplify this expression as this form keeps coming in our circuits class (superposition of two sinusoids with same frequency but different phase and amplitude). I feel using complex numbers or vectors gives more insight into how the sinusoids mix. I'm into engg and generally scared of trying new things in math, so I seek your help..

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I know one way to simplify this by expanding both sin terms using the formula sin(A+B) and get it into the form $a \cos x + b \sin x$. Then use the formula :

Answer 1

You can use that $$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$$

Answer 2

The function $\,f(x) := 2 \sin(x + \pi/5) + 3 \sin(x + \pi/7),\,$ is a sine wave. Thus for some constants $\,a,b,c,\,$ $\,f(x) = b \sin(x + a) + c \cos(x + a).\,$ Let $\,a = (\frac{\pi}5 + \frac{\pi}7)/2 = \pi \frac{6}{35}.\,$ To solve for $\,c\,$ let $\,x=-a,\,$ so that $\,f(-a) = c = 2\sin(\frac{\pi}{35}) + 3 \sin(-\frac{\pi}{35}) = -\sin(\frac{\pi}{ 35})\,$ and to solve for $\,b\,$ let $\,x = \frac{\pi}2 - a ,\,$ so that $f(\frac{\pi}2 - a) = b = 2 \sin(\pi\, \frac{37}{70}) + 3 \sin(\pi\,\frac{33}{70}) = 5 \cos(\frac{\pi}{35}).\,$