Sine, cosine, and polar form of complex numbers

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We define two trigonometric functions, namely the sine and cosine functions, based on the geometric interpretation of complex numbers. Using these functions, we define an alternative representation of complex numbers called the polar form.



Definition (Sine and cosine)

Let \(u \in \mathbb{C}\) such that \(|u| = 1\) and \(\theta \in \arg(u)\). We define \(\sin: \mathbb{R} \to \mathbb{R}\) and \(\cos: \mathbb{R} \to \mathbb{R}\) by 

\[\begin{eqnarray} \cos(\theta) &=& \Re(u),\\ \sin(\theta) &=& \Im(u). \end{eqnarray}\]

Thus, we have \(u = \cos\theta + i\sin\theta\).


Example. In particular, we have

\[\begin{eqnarray} \cos(0) &=& 1,\\ \cos(\pi/2) &=& 0,\\ \cos(\pi) &=& -1,\\ \sin(0) &=& 0,\\ \sin(\pi/2) &=& 1,\\ \sin(\pi) &=& 0. \end{eqnarray}\]

Also, note that

\[\begin{eqnarray} \cos(-\theta) &=& \cos\theta,\\ \sin(-\theta) &=& -\sin\theta. \end{eqnarray}\]

Draw the figure of the unit circle to confirm these values. □

Based on this geometric definition, we prove the following theorem geometrically.

Theorem (Compound angle formulae)

Suppose that \(\alpha, \beta \in \mathbb{R}\). Then

  1. \(\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha \sin\beta\),
  2. \(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha \sin\beta\).
Proof. Let \(u = \cos\alpha + i\sin\alpha\) and \(v = \cos\beta + i\sin\beta\). Note that, by the definition of \(\cos\) and \(\sin\), \(|u| = |v| = 1\). Thus multiplying \(v\) by \(u\) has the effect of rotating \(v\) by \(\text{Arg}(u) = \alpha\). Thus \(\text{Arg}(uv) = \alpha + \beta\). Hence
\[\begin{eqnarray} \cos(\alpha + \beta) + i\sin(\alpha + \beta)&=&uv\\ &=& (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta)\\ &=& (\cos\alpha\cos\beta - \sin\alpha \sin\beta)\nonumber\\ && + i(\sin\alpha\cos\beta + \cos\alpha \sin\beta). \end{eqnarray}\]
By comparing the real and imaginary parts, we obtain the desired results. ■

Theorem (De Moivre)

Let \(\theta \in \mathbb{R}\) and \(n\in\mathbb{N}\). Then

\[(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta).\]

Proof. We prove this by mathematical induction.

The equality trivially holds for \(n = 1\).

Suppose the equality holds for \(n = k\). Then

\[(\cos\theta + i \sin\theta)^{k+1} = (\cos\theta + i \sin\theta)^k(\cos\theta + i \sin\theta)\]

(by the induction hypothesis)

\[\begin{eqnarray} &=& (\cos(k\theta) + i \sin(k\theta))(\cos\theta + i \sin\theta)\\ &=& [\cos(k\theta)\cos\theta -\sin(k\theta)\sin\theta]\nonumber\\ && + i[\sin(k\theta)\cos\theta + \cos(k\theta)\sin\theta] \end{eqnarray}\]

(by the compound angle formulae)

\[= \cos[(k+1)\theta] + i \sin[(k+1)\theta].\]

■

For any \(z\in\mathbb{C}, z\neq 0\), we have the ``decomposition'' \(z = |z|\cdot\frac{z}{|z|}\). Since \(\left|\frac{z}{|z|}\right| = 1\), there is a \(\theta \in \mathbb{R}\) such that 

\[\frac{z}{|z|} = \cos\theta + i\sin\theta.\]

Definition (Polar form)

Let \(z\in\mathbb{C}, z\neq 0\). Let \(r = |z|\) and choose a \(\theta \in \arg(z)\). The polar form of \(z\) is its expression as \(r(\cos\theta + i\sin\theta)\).

Example. Let \(z = \sqrt{6} + \sqrt{2}i\). Then, \(|z| = 2\sqrt{2}\), so

\[z = 2\sqrt{2}\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = 2\sqrt{2}\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right).\]

Example. Let \(z = r(\cos\alpha + i\sin\alpha)\) and \(w = s(\cos\beta + i\sin\beta)\). Then

\[\begin{eqnarray} zw &=& rs[\cos(\alpha + \beta) + i\sin(\alpha+\beta)],\\ 1/z &=& (1/r)[\cos(-\alpha) + i\sin(-\alpha)]\\ &=&(1/r)[\cos\alpha - i\sin\alpha]\\ &=& \bar{z}/|z|^2,\\ z/w &=& (r/s)[\cos(\alpha - \beta) + i\sin(\alpha - \beta)]. \end{eqnarray}\]

Draw figures! □

Definition (Periodic functions)

Let \(f: \mathbb{R} \to X\) be a function where \(X\) is any set. We say \(f\) is periodic if there exists a positive \(p \in\mathbb{R}\) such that \(f(x + p) = f(x)\) for all \(x \in \mathbb{R}\), and \(p\) is called a period of the function. If \(f\) is periodic, and there exists a smallest period, the smallest period is called the fundamental period.

Example. Both \(\sin\) and \(\cos\) are periodic functions with fundamental period of \(2\pi\). □

Example. The tangent function is defined as

\[\tan x = \frac{\sin x}{\cos x}.\]

But this is not a function from \(\mathbb{R}\) to \(\mathbb{R}\) because it ``explodes'' (or ``has a singularity'') at odd integer multiples of \(\pi/2\) where \(\cos x = 0\). We invent formal symbols \(+\infty\) and \(-\infty\) to make \(\tan\) a function, \(\tan: \mathbb{R} \to \mathbb{R}\cup \{\pm \infty\}\). Alternatively, we can remove odd multiples of \(\pi/2\) from the domain. In any case, \(\tan\) can be shown to have the fundamental period of \(\pi\). (Draw the graph!) □


Definition (Even and odd functions)

Let \(f: \mathbb{R} \to \mathbb{R}\) be a function. We say \(f\) is even if \(f(x) = f(-x)\) for every \(x\in \mathbb{R}\). We say \(f\) is odd if \(f(-x) = -f(x)\) for every \(x\in\mathbb{R}\). 

Example. \(\sin\) is an odd function. \(\cos\) is an even function. □





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