<s>[INST] Simplify $5(3 \sin(x) + \sqrt3 \cos(x))$ So Wolfram tells me I can reach $10 \sqrt3 \sin(x + π/6)$ from this expression, but I cant grasp how to do it. Any help is appreciated.
 [/INST] We know 
$$\sin (x + a) = \sin x \cos a + \sin a \cos x$$ 
While we have 
$$15 \sin x + 5 \sqrt 3 \cos x \tag {*}$$
Therefore if we can find an numbers $a, C$ such that $15 = C \cos a$ and $5\sqrt3 = C\sin a$, we can rewrite $(*)$ as 
$$C \cos a \sin x + C \sin a \cos x = C\sin (x+a)$$
To find appropriate $C$ and $a$, notice that we have a system of two equations
$\left\{ 
\begin{array}{c}
C \cos a = 15 \\ 
C \sin a = 5 \sqrt 3 \\
\end{array}
\right.$
Square both sides of both equations and add the resulting equations. We get:
$C^2 \sin ^2 a + C^2 \cos ^2 a = 15^2 + 25 \cdot 3$
$C^2(\sin^2 a + \cos^2 a) = 300$
$C^2 = 300$
$C = 10 \sqrt 3$
Now subsitute in one of the equations
$10 \sqrt 3 \sin a = 5 \sqrt 3$
$\sin a = \dfrac 12$
$a = \dfrac {\pi}{6}$
Another possible value for $a$ is $\dfrac {5 \pi}{6}$, but only $\dfrac {\pi}{6}$ satisfies both equations.
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