TITLE: Expressing a number in $\sqrt a/b$ form
QUESTION [5 upvotes]: Express the number $\sqrt3 \sin(10^\circ) +\dfrac38\tan(10^\circ)$ in the form $\dfrac{\sqrt a} b$, where $a$ and $b$ are integers.

I am sure that trigonometric formulas must be used here, but I cannot see how. I will also be appreciative if someone gives me any hints.

REPLY [2 votes]: I think it should $\frac34$ instead of $\frac38$
$$\sin(30^\circ-10^\circ)=\sin20^\circ$$
$$\implies \sin30^\circ\cos10^\circ-\cos30^\circ\sin10^\circ=2\sin10^\circ\cos10^\circ$$
$$\implies \frac12\cos10^\circ-\frac{\sqrt3}2\sin10^\circ=2\sin10^\circ\cos10^\circ$$
$$\implies \frac12-\frac{\sqrt3}2\tan10^\circ=2\sin10^\circ$$ (Dividing by $\cos10^\circ\ne0$)
$$\implies \frac{\sqrt3}2\tan10^\circ+2\sin10^\circ=\frac12$$
Multiply by $\frac{\sqrt3}2$