Question

Simplify asin(x/2)cos(x/2), is the answer sin2(x/2) ?

Answer 1

asin?

Answer 2

$$\sin2\theta=2\sin\theta\cos\theta\implies\sin\theta\cos\theta=\frac12\sin2\theta$$Here, we have \(a\sin\frac{x}2\cos\frac{x}2=\frac{a}2\sin x\).

Answer 3

so... not sin2(x/2)

Answer 4

Jeez that was a roundabout derivation @Mertsj :-p

Answer 5

@oldrin.bataku Sorry. I'll take it down. Didn't mean to offend you.

Answer 6

@Mertsj huh? no offense here! I just thought it was longer than needed :-p

Answer 7

We both came to the same result :-)

Answer 8

ok, sorry this is such a late response, but why is it (a/2)sinx rather than (a/2)sin2x.

Answer 9

You need to ask the genius @oldrin.bataku

Answer 10

@Cutiepo0
$$\sin2\theta=2\sin\theta\cos\theta\implies\sin\theta\cos\theta=\frac12\sin2\theta$$Understand so far? Now notice our expression can be treated as follows:$$a\sin\frac{x}2\cos\frac{x}2=a\left(\sin\frac{x}2\cos\frac{x}2\right)$$With \(\theta=x/2\), we observe that we apply our identity to yield:$$\sin\frac{x}2\cos\frac{x}2=\frac12\sin2\left(\frac{x}2\right)=\frac12\sin x$$Does that make sense so far? Now we put it all together:$$a\left(\sin\frac{x}2\cos\frac{x}2\right)=a\left(\frac12\sin x\right)=\frac12a\sin x$$