]> Exercise 2.10

## Exercise 2.10

Derive the formulae for $sin ⁡ x$ and $cos ⁡ x$ in terms of $exp ⁡ i x$ and $exp ⁡ ( − i x )$ , that follows from equation (A) above.

Solution:

$exp ⁡ i x = cos ⁡ x + i sin ⁡ x$

and it holds for all $x$ positive or negative, so that we also have

$exp ⁡ ( − i x ) = cos ⁡ x − i sin ⁡ x$

where we have used the fact that cosine is an even function and sine an odd one.

We can solve these two equations for $cos ⁡ x$ and $sin ⁡ x$

$cos ⁡ x = exp ⁡ i x + exp ⁡ ( − i x ) 2 sin ⁡ x = exp ⁡ i x − exp ⁡ ( − i x ) 2 i$