Computing sine and cosine of complex arguments with only real functions

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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 np

def 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 + 4j
mysin = 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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The post Computing sine and cosine of complex arguments with only real functions first appeared on John D. Cook.

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