Rewrite the following expression in the form $A \sin(x+C)$ $$4 \sin x + 4\sqrt{3} \cos x$$
This is what I have so far, and I'm not even sure it's the right approach. I just dont understand this concept as a whole:
$$A \cos(c)\sin(x) + A\sin(c)\cos(x)$$
$$A\cos (c)=4$$
$$A\sin (c)=4\sqrt{3}$$
$$\sin^2(x) + \cos^2(x) = 1$$ ??
Hint: As you wrote above, $\sin(x+c) = \cos(c) \sin(x) + \sin(c) \cos(x)$. Imagine that you could find some $c$ so that $\cos(c) = \frac{1}{2}$ and $\sin(c) = \frac{\sqrt{3}}{2}$, you would then have $$\sin(x + c) = \frac{1}{2} \sin(x) + \frac{\sqrt{3}}{2} \cos(x) = \frac{1}{2} \big[\sin(x) + \sqrt{3} \cos(x)\big].$$
rewrite your term in the form $\sqrt{4^2+4^2\cdot 3}\left(\frac{4}{\sqrt{4^2+4^2\cdot3}}\sin(x)+\frac{4\sqrt{3}}{\sqrt{4^2+4^2\cdot3}}\cos(x)\right)$ now we have $\cos(\phi)=\frac{4}{\sqrt{4^4+4^2\cdot 3}}$ and $\sin(\phi)=\frac{4\sqrt{3}}{\sqrt{4^2+4^2\cdot3}}$
