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.
 A: $a \sin x + b\cos x$
let $\phi = \arctan \frac{b}{a}$
$\sin {\phi} = \frac {b}{\sqrt {a^2+b^2}}\\
\cos {\phi} = \frac {a}{\sqrt {a^2+b^2}}$
$a \sin x + b\cos x = \sqrt {a^2 + b^2} (\cos\phi \sin x + \sin\phi\cos x) = \sqrt {a^2 + b^2} \sin (x+\phi)$
A: 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.
A: its $$10\sqrt3\left(\frac{\sqrt3}{2}\sin{x}+\frac{1}{2}\cos{x}\right)=10\sqrt3\left(\sin60^{\circ}\sin{x}+\cos60^{\circ}\cos{x}\right)=10\sqrt3\cos\left(60^{\circ}-x\right)$$
In your case it's
$$10\sqrt3\left(\cos30^{\circ}\sin{x}+\sin30^{\circ}\cos{x}\right)$$
A: Given these identities:
$$\frac{1}{2}\sqrt{3} = \cos(\frac{\pi}{6})$$
$$\frac{1}{2} = \sin(\frac{\pi}{6})$$
$$\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha)$$
We get:
$$5(3\sin(x) + \sqrt{3}\cos(x)) = $$
$$5\sqrt{3}(\sqrt{3}\sin(x) + \cos(x)) =$$
$$10\sqrt{3}(\cos(\frac{\pi}{6})\sin(x) + \frac{1}{2}\cos(x)) =$$
$$10\sqrt{3}(\cos(\frac{\pi}{6})\sin(x) + \sin(\frac{\pi}{6})\cos(x)) =$$
$$10\sqrt{3}\sin(x + \frac{\pi}{6})$$
A: $$5(3sin(x)+\sqrt{3}cos(x))=10\sqrt{3}(\frac{\sqrt{3}}{2}sin(x)+\frac{1}{2}cos(x))=10\sqrt{3}(sin(x+\frac{\pi}{6}))$$
