# How do you prove Sin(x/3)cos(x/3) = (1/2)sin(2x/3)?

Apr 27, 2018

We substitute $\theta = \frac{x}{3}$ in the $\sin \left(2 \theta\right) = 2 \sin \theta \cos \theta$
and we're home.

#### Explanation:

The double angle formula for sine is

$\sin \left(2 \theta\right) = 2 \sin \theta \cos \theta$

That's an old saw your teacher will generally accept without proof.

Setting $\theta = \frac{x}{3}$,

$\sin \left(2 \cdot \frac{x}{3}\right) = 2 \sin \left(\frac{x}{3}\right) \cos \left(\frac{x}{3}\right)$

 1 /2 sin((2x}/3) = sin (x/3) cos(x/3) quad sqrt