Complex Trigonometric Functions¶

Definition¶

Let $z\in\mathbb{C}$ $C$ . The complex sine function $\sin z$ and the complex cosine function $\cos z$ are defined as

\begin{aligned} \sin z&=\sum_{n=0}^{\infty} \frac{(-1)^{n} z^{2 n+1}}{(2 n+1) !}\\ &=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\cdots,\\ \cos z&=\sum_{n=0}^{\infty} \frac{(-1)^{n} z^{2 n}}{(2 n) !}\\ &=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\cdots, \end{aligned}

where the powers $z^n$ are defined by repeated complex multiplication and can be evaluated in polar form by De Moivre's formula.

Interpretation¶

The complex sine and cosine extend the real trigonometric functions to complex inputs. A complex number has no point on the unit circle to read the values from, so the definition uses infinite series instead: each power $z^n$ is computed by complex multiplication, and the terms are added up.

The sine series contains only odd powers and the cosine series only even powers, with alternating signs and factorials in the denominators. Both series converge for every $z\in\mathbb{C}$ $C$ . For a real input $x$ the two sums are the familiar $\sin x$ and $\cos x$ , so nothing changes on the real axis: the definitions extend sine and cosine rather than replace them.

Off the real axis they behave differently, and the bound $|\sin x|\le1$ from the unit circle is lost. On the imaginary axis, $(\mathrm{i}y)^{2n}=(-1)^ny^{2n}$ cancels the alternating signs, so $\cos(\mathrm{i}y)=1+\frac{y^2}{2!}+\frac{y^4}{4!}+\cdots$ for real $y$ . This is larger than $1$ for every $y\neq0$ and grows without bound; the Examples show the same for $\sin\mathrm{i}$ .

A common slip concerns Euler's formula. The identity $\mathrm{e}^{\mathrm{i}z}=\cos z+\mathrm{i}\sin z$ holds for every complex $z$ , but for non-real $z$ the numbers $\cos z$ and $\sin z$ are in general not real, so they are not the real and imaginary parts of $\mathrm{e}^{\mathrm{i}z}$ . For $z=\mathrm{i}$ $i$ , $\mathrm{e}^{\mathrm{i}\cdot\mathrm{i}}=\mathrm{e}^{-1}$ is real, while $\sin\mathrm{i}\approx1.1752\,\mathrm{i}\neq0$ $0$ .

Why the real sine and cosine are given by exactly these series is explained later by Taylor series.

Key Results¶

The complex sine and cosine functions remain periodic with period $2\pi$ .
The algebraic identities of real trigonometry, such as $\sin^2z+\cos^2z=1$ and the addition formulas, continue to hold for complex arguments.
By Euler's formula, $\cos z=\dfrac{\mathrm{e}^{\mathrm{i}z}+\mathrm{e}^{-\mathrm{i}z}}{2}$ and $\sin z=\dfrac{\mathrm{e}^{\mathrm{i}z}-\mathrm{e}^{-\mathrm{i}z}}{2\mathrm{i}}$ $\frac{e ^{i z} - e ^{- i z}}{2 i}$ .
Unlike the real functions, $\sin z$ and $\cos z$ are unbounded on $\mathbb{C}$ : $|\sin z|$ can be larger than $1$ .

Examples¶

Since $\mathrm{i}^3=-\mathrm{i}$ and $\mathrm{i}^5=\mathrm{i}$ $i$ , the signs in the series for $\sin\mathrm{i}$ all turn positive:

\begin{aligned} \sin\mathrm{i}&=\mathrm{i}+\frac{\mathrm{i}}{3!}+\frac{\mathrm{i}}{5!}+\cdots\\ &=\mathrm{i}\left(1+\frac{1}{3!}+\frac{1}{5!}+\cdots\right)\\ &=\mathrm{i}\,\frac{\mathrm{e}-\mathrm{e}^{-1}}{2}\approx1.1752\,\mathrm{i}. \end{aligned}

So $|\sin\mathrm{i}|\approx1.1752>1$ $1$ , which no real sine value can reach. The same value follows from Euler's form: $\sin\mathrm{i}=\dfrac{\mathrm{e}^{-1}-\mathrm{e}}{2\mathrm{i}}=\mathrm{i}\,\dfrac{\mathrm{e}-\mathrm{e}^{-1}}{2}$ $i \frac{e - e ^{- 1}}{2}$ .

Prerequisites¶

Taylor Series — later development in Math III; the defining series is given above.
De Moivre's Formula
Periodic Function

Sources¶

Math II: Linear Algebra · Week 1

Symbols: complex numbers, element of, infinity, pi, sum

Last updated: 20 October 2026 03:02