Question

How do you derive the multiple angles formula?

Answer 1

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:

Multiple Angle Formulas