]> 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:

Equation (A) reads

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