The circular exponential and trigonometric functions

From the complex exponential function, we can define a “circular exponential” function, which “wraps” the real line around the trigonometric circle, and makes it possible to rigorously define the cosine and sine trigonometric functions, which extend to the entire complex plane, and to demonstrate their elementary properties.

Introduction: defining trigonometric functions

In “The trigonometric circle” we discussed the trigonometric functions cosine and sine, which are rigorously defined geometrically for (oriented) angles. Their definition as functions \(\cos : \mathbb R\to \mathbb R\) and $\sin:\mathbb R\to \mathbb R$ is more delicate, and for the moment we have only given an intuitive explanation, corresponding to the “winding” of the real line onto the trigonometric circle.

The cosine and sine functions of a real variable are called circular functions, and we will define them rigorously here thanks to the function we will call the ‘circular’ exponential, which we will introduce from the complex exponential. The usual properties of $\cos$ and $\sin$ will be derived from this function, and this will be an opportunity to introduce the trigonometric representation and notation of complex numbers.

1 The “circular” exponential

1.1 Definition based on the complex exponential

The analytic function $\exp:\mathbb C\to \mathbb C^*$ is defined for any complex number $z\in\mathbb C$ as the sum of the convergent series $\sum \dfrac{z^n}{n!}$. This function is in fact an extension of the real exponential function $\exp:\mathbb R\to \mathbb R_+^*$ (see Analytic functions and complex exponential), and if we consider its restriction to the set $i\mathbb R={z\in\mathbb C : \exists x\in\mathbb R,\ z=ix}$ of pure imaginary numbers, we define a new function, which we call a “circular” exponential.
To be more precise, the function we are referring to is the application $e$ defined on the set $\mathbb R$, and with values in $S^1={z\in\mathbb C : |z|=1}$ of complex numbers with module $1$, by $e(t)=\exp(it)$. We noted when studying the complex exponential that for any pure imaginary number $z=x+iy$, we have $|\exp(z)|=1$. The circle $S^1$ being none other than the trigonometric circle, the function $e$ is none other than the “winding” of the real line on this circle, from which we have evoked a possible definition of the cosine and sine functions.

1.2. Properties of $e:t\mapsto \exp(it)$

Before deriving a rigorous definition of cosine and sine from this function, we will explain a few elementary properties, drawn from the properties of the complex exponential.

The trigonometric circle $S^1$ is “stable” under multiplication: if $z,w\in S^1$ are two complex numbers with modulus $1$, their product $z.w$ also has modulus $1$. As the complex exponential is a “group homomorphism” from $(\mathbb C,+)$ into $(\mathbb C^*,\times)$, this means that the circular exponential transforms the addition of real numbers into the multiplication of complex numbers of modulus $1$. In other words, if $s,t$ are two real numbers, we have $e(s+t)=\exp(i(s+t))=\exp(is+it)=\exp(is)\times\exp(it)=e(s)\times e(t)$ : the function $e$ is itself a group homomorphism, from the group $(\mathbb R,+)$ into the group $(S^1,\times)$.

As a function from $\mathbb R$ into $\mathbb C$, the application $e$ is a function from $\mathbb R$ into $\mathbb R^2$ and as such, it is derivable as an analytic function. In particular, it is continuous, and an expression for its derivative is then given from its derivative series, and since we have $e(t)=\sum_{n=0}^{+\infty} \dfrac{(it)^n}{n!}=\sum_{n=0}^{+\infty} \dfrac{i^n}{n!} t^n$, this series gives us the value of $e'(t)$, i.e. $\sum_{n=0}^{+\infty} (n+1)\dfrac{i^{n+1}}{(n+1)!} t^n=i\sum_{n=0}^{+\infty} \dfrac{i^n}{n!} t^n=ie(t)$. This formula will be useful for the derivation of the real cosine and sine.

The circular exponential function “wraps” the real line around the trigonometric circle. Using the theory of rectifiable arcs, it can be shown that the length of the arc bounded by the point $I$ and the point $\exp(it)$ is precisely $t$.

2 Definition and properties of cosine and sine

2.1 A rigorous definition of the $\cos$ and $\sin$ functions

Once the circular exponential has been defined as the “winding” of the real line onto the trigonometric circle, the definition of the usual circular functions is straightforward. We define the cosine of any real number $t$ as the real part of $e(t)=\exp(it)$, and the sine of $t$ as the imaginary part of $e(t)$. Composing the function $e:\mathbb R\to \mathbb C$ by the functions real part $Re:\mathbb C\to \mathbb R,\ x+iy\mapsto x$ and imaginary part $Im: x+iy\in\mathbb C\mapsto y\in \mathbb R$ – which are just the two projections of the Euclidean plane, we obtain two functions, the cosine $\cos:x\in\mathbb R\mapsto Re(\exp(it)) \in\mathbb R$ and the sine $\sin:x\in\mathbb R\mapsto Im(\exp(it))\in\mathbb R$.

From the definition of $e:\mathbb R\to \mathbb C$ as an analytic function and the properties of the real part and the imaginary part of the sum of a series of complex numbers, we can then give a description of the cosine and the sine as sums of series. Indeed, since for any real number $t$ we have $e(t)=\exp(it)=\sum_{n=0}^{+\infty} \dfrac{(it)^n}{n!}$, using the fact that $i^{2n}=(-1)^n$ and $i^{2n+1}=(-1)^ni$ for any natural number $n$, we get $$\cos(t)=Re(e(t))=\sum_{n=0}^{+\infty} (-1)^n\dfrac{t^{2n}}{(2n)! }$$ and $$\sin(t)=Im(e(t))=\sum_{n=0}^{+\infty} (-1)^n\dfrac{t^{2n+1}}{(2n+1)!},$$ which means that the functions $\cos$ and $\sin$ are real analytic functions.

2.2 Recovering the properties of the trigonometric functions

From the properties of the circular exponential function, it is possible to rigorously establish the elementary properties of the real $\cos$ and $\sin$ functions. To do this, all we have to do is represent the function $e$ from the functions $\cos$ and $\sin$: in fact, for any real number $t$, we can now write $e(t)=\exp(it)=\cos t+i\sin t$, by definition! For example, for all real numbers $s,t$ we have $$\cos(s+t)+i\sin(s+t)=\exp(i(s+t))=\exp(is)\exp(it)=(\cos s+i\sin s).(\cos t+i\sin t).$$ Taking the real part of the two extreme members gives $cos(s+t)=\cos s\cos t-\sin s\sin t$. Taking the imaginary parts this time, we also get $\sin(s+t)=\sin s \cos t+\sin t\cos s$: we recover the fundamental trigonometric identities.

We have already differentiated the function $e$, and we can now write, for any $t\in\mathbb R$, $$e'(t)=ie(t)=i\exp(it)=i(\cos t+i\sin t)=-\sin t+i\cos t.$$ Now, the derivative of a function from $\mathbb R$ into $\mathbb R^2$ is obtained by differentiating each coordinate. Since the function $e$ can be represented as $t\in\mathbb R\mapsto (\cos t,\sin t)$, we can directly derive the expression for the derivatives of the functions $\cos t$ and $\sin t$. In fact, we have $e'(t)=(\cos’t,\sin’t)$, from which $\cos’ t=-\sin t$ and $\sin’t=\cos t$, which can be found by deriving the series representing $\cos$ and $\sin$ as analytic functions! Note that the norm of $e'(t)$ is $||e'(t)||=\sqrt{(-\sin t)^2+(\cos t)^2}=1$, so the length of the arc determined by $I$ and $\exp(it)$ is $\int_0^t ||e'(x)||\ dx=\int_0^t dx=t$.

3 The complex circular functions

3.1 Expressing $\cos t$ and $\sin t$ as functions of $\exp(it)$

From the expression $\exp(it)=\cos t+i\sin t$, we can give expressions for $\cos t$ and $\sin t$ as functions of $\exp(it)$. For any complex number $z=x+iy$, we know that $x=Re(z)=\frac 1 2 (z+\overline z)$ and $y=Im(z)=\frac 1 {2i} (z-\overline z)$. Now, the conjugate $\overline z=x-iy$ of a complex number $z=x+it$ of the form $z=\exp(it)$ is $\overline z=\exp(-it)$ : it follows that we have $$\cos t=Re(\exp(it))=\dfrac{\exp(it)+\exp(-it)}{2}$$ and $$\sin t=Im(\exp(it))=\dfrac{\exp(it)-\exp(-it)}{2i}.$$

3.2 Extending cosine and sine to the whole set $\mathbb C$

In this expression, there is nothing to prevent the real number $t$ from being replaced by any complex number $z$ : we would thus define two functions from $\mathbb C$ into $\mathbb C$, still called (complex) “cosine” and “sine”, and defined by $$\cos z=\dfrac{\exp(iz)+\exp(-iz)}{2}$$ and $$\sin z=\dfrac{\exp(iz)-\exp(-iz)}{2i}.$$ Caution: in the general case, the relations to the real and imaginary parts of $z$ are more complicated.
These functions could also be defined from the power series expansions written above, which extend to convergent series throughout the complex plane, namely $$\cos(z)= \sum_{n=0}^{+\infty} (-1)^n\dfrac{z^{2n}}{(2n)! }$$ and $$\sin(z)=\sum_{n=0}^{+\infty} (-1)^n\dfrac{z^{2n+1}}{(2n+1)!}$$ for any complex number $z$. We can therefore work out in two different ways that the functions $\cos,\sin:\mathbb C\to \mathbb C$ are analytic.

Such functions, like the complex exponential, which are analytic and defined on the whole set $\mathbb C$, are said to be entire (or integral), and Liouville’s theorem tells us that a bounded entire function is constant. Now the cosine and sine functions are not constant, so they are not bounded! In other words, while the cosine and sine of a real number always take values between $-1$ and $1$ (these are the coordinates of a point on the trigonometric circle), the cosine and sine of a complex number can take arbitrarily large values!


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