Exercise 2.10

]> Exercise 2.10

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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 $\sin x$

\begin{array}{l} \cos x = \frac{\exp i x + \exp (- i x)}{2} \\ \sin x = \frac{\exp i x - \exp (- i x)}{2 i} \end{array}