How do you derive the multiple angles formula?

Question

5707 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-04-12T16:35:17

One approach is to use repeated applications of the sum formulas.

$trig(2a) = trig (a+a)$

$trig(3a) = trig (2a+a)$

$trig(4a) = trig (3a+a)$ (or $=trig(2a+2a)$ )

Another approach is to rewrite using Euler's Formula: $e^(i n x) = cos(nx)+i sin(nx)$

So $sin(nx) = (e^( i n x) - e^( - i n x))/(2 i) = ((e^( i x))^n - (e^( - i x))^n)/(2 i)$

$= ((cosx+isinx)^n - (cosx-isinx)^n)/(2 i)$ .

Now expand using the binomial theorem, and simplify.

Here's an outside link with the details: