How do you prove $sin (\frac{x}{3}) cos (\frac{x}{3}) = (\frac{1}{2}) sin (2 \frac{x}{3})$ $Sin(x/3)cos(x/3) = (1/2)sin(2x/3)$ ?

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Answer 1 · 2018-04-27T15:50:34

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

The double angle formula for sine is

$sin (2 θ) = 2 sin θ cos θ$ $sin(2 theta ) = 2 sin theta cos theta$

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

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

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

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

How do you prove Sin(x/3)cos(x/3) = (1/2)sin(2x/3)sin(x3)cos(x3)=(12)sin(2x3)Sin(x/3)cos(x/3) = (1/2)sin(2x/3)?