Home Statistics and Visualization The Endeavour Computing sine and cosine of complex arguments with only real functions

Computing sine and cosine of complex arguments with only real functions

The Endeavour 2026-03-27

Suppose you have a calculator or math library that only handles real arguments but you need to evaluate sin(3 + 4i). What do you do?

If you’re using Python, for example, and you don’t have NumPy installed, you can use the built-in math library, but it will not accept complex inputs.

>>> import math>>> math.sin(3 + 4j)Traceback (most recent call last):File "<stdin>", line 1, in <module>TypeError: must be real number, not complex

You can use the following identities to calculate sine and cosine for complex arguments using only real functions.

\begin{align*} \sin(x + iy) &= \sin x \cosh y + i \cos x \sinh y \\ \cos(x + iy) &= \cos x \cosh y - i \sin x \sinh y \\ \end{align*}

The proof is very simple: just use the addition formulas for sine and cosine, and the following identities.

\begin{align*} \sin iz &= i \sinh z \\ \cos iz &= \cosh z \end{align*}

The following code implements sine and cosine for complex arguments using only the built-in Python functions that accept real arguments. It then tests these against the NumPy versions that accept complex arguments.

from math import *import numpy as npdef complex_sin(z):    x, y = z.real, z.imag    return sin(x)*cosh(y) + 1j*cos(x)*sinh(y)def complex_cos(z):    x, y = z.real, z.imag    return cos(x)*cosh(y) - 1j*sin(x)*sinh(y)z = 3 + 4jmysin = complex_sin(z)mycos = complex_cos(z)npsin = np.sin(z)npcos = np.cos(z)assert(abs(mysin - npsin) < 1e-14)assert(abs(mycos - npcos) < 1e-14)

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