The Trigonometric Functions

Section 3.4 The Trigonometric Functions

The trigonometric functions arise from geometric considerations. It was Euler who discovered the relation between the trigonometric functions and the exponential function as encoded in the Euler formula

\begin{equation*} E(ix)=\cos (x) + i \sin (x). \end{equation*}

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\(E(ix)\) is defined in terms of the power series expansion

\begin{equation*} \sum_{n=0}^{\infty} \frac{(i x)^{n}}{n!}. \end{equation*}

For real valued \(x\text{,}\) splitting the above series into its real and imaginary parts, we obtain

\begin{equation*} \cos x = \sum_{k=0}^{\infty}\frac{(-1)^{k} x^{2k}}{(2k)!}, \quad \sin x= \sum_{k=0}^{\infty}\frac{(-1)^{k} x^{2k+1}}{(2k+1)!}. \end{equation*}

But the series converges for all complex valued \(x\text{,}\) so we also define \(\cos x\) and \(\sin x\) for complex valued \(x\) using the above series.

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Note that the definitions for these trigonometric functions are in purely analytic means, thus these trigonometric functions as defined this way do not directly have relations with the ones defined geometrically. Rudin’s development follows this approach and does not use any of the properties as given by the geometric approach. For instance, he sets out to prove that \(E(x)\) has a purely imaginary period, labeled as \(2\pi i\text{,}\) and use this to show that both \(\cos x\) and \(\sin x\) also have \(2\pi i\) as their period; but the \(2\pi\) here does not have a direct relation with the angle interpretation in geometry. This kind of treatment is fine for a rigorous development of calculus, but should not be taken as a discouragement from relating to the geometric approach. In fact it is much more productive to fully use the geometric interpretation; one just needs to be ware of the places where a certain geometric properties play a crucial role and how those arguments can be replaced by purely analytical ones.

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