Complex numbers: Complex functions
Complex trigonometric functions
Once you have the complex exponential functie at your disposal, you can also define the complex trigonometric functions \(\cos\) and \(\sin\) in a simple way.
\[\cos(z)=\frac{e^{z\,\mathrm{i}}+e^{-z\,\mathrm{i}}}{2}\qquad\text{and}\qquad\sin(z)=\frac{e^{z\,\mathrm{i}}-e^{-z\,\mathrm{i}}}{2\mathrm{i}}\]
The periodicity of the complex exponential function with period \(2\,\pi\,\ii\) immediately leads to the periodicity of the complex trigonometric functions.
Periodicity The complex trigonometric functions \(\cos\) and \(\sin\) are periodic functions with period \(2\,\pi\).
For the complex exponential functions holds \[e^{z+2\,\pi\,\ii}=e^z\] The periodicity of the cosine follows from the following calculation: \[\begin{aligned}\cos(z+2\pi)&=\frac{1}{2}\left(e^{(z+2\pi)\cdot\ii}+e^{-(z+2\pi)\cdot\ii}\right)\\ &\phantom{rstuvwxyz}\color{blue}{\text{definition}}\\ &=\frac{1}{2}\left(\e^{z\cdot\ii}+\e^{-z\cdot\ii}\right) \\ &\phantom{rstuvwxyz}\color{blue}{\text{periodicity of }\exp}\\ &=\cos(z) \\ &\phantom{rstuvwxyz} \color{blue}{\text{definition}}\end{aligned}\] The proof that the sine is periodic with period #2\pi# is the same.
Also, the following well-known formula is valid for all complex numbers.
\[\cos^2(z)+\sin^2(z)=1\qquad\text{for any complex number }z\]
For the maths enthusiast, we give the proof of the theorem that \[\cos^2(z)+\sin^2(z)=1\qquad\text{for any complex number }z\] This proof is in fact no more than the use of the definitions of trigonometric functions and expansion of the squares. \[\begin{aligned}\cos^2(z)+\sin^2(z) &= \left(\frac{e^{z\,\ii}+e^{-z\,\ii}}{2}\right)^2+ \left(\frac{e^{z\,\ii}-e^{-z\,\ii}}{2\ii}\right)^2\\ & \phantom{uvwxyz}\color{blue}{\text{definitions}} \\ &= \frac{\bigl(e^{z\,\ii}\bigr)^2+2\cdot e^{z\,\ii}\cdot e^{-z\,\ii}+\bigl(e^{-z\,\ii}\bigr)^2}{4}\\ &\phantom{=}\quad {}+ \frac{\bigl(e^{z\,\ii}\bigr)^2-2\cdot e^{z\,\ii}\cdot e^{-z\,\ii}+\bigl(e^{-z\,\ii}\bigr)^2}{4\ii^2}\\ & \phantom{uvwxyz}\color{blue}{\text{sum and difference formula of squares}} \\ &= \frac{e^{2z\,\ii}+2+e^{-2z\,\ii}}{4} - \frac{e^{2z\,\ii} - 2+e^{-2z\,\ii}}{4}\\ & \phantom{uvwxyz}\color{blue}{\text{calculation rules for powers and }\ii^2=-1} \\ &= \frac{e^{2z\,\ii}+2+e^{-2z\,\ii}-e^{2z\,\ii} + 2-e^{-2z\,\ii}}{4}\\ & \phantom{uvwxyz}\color{blue}{\text{everything under one common denominator}}\\ &= \frac{4}{4}= 1 \\ & \phantom{uvwxyz}\color{blue}{\text{simplification}}\end{aligned}\]
Also the double-angle formulas, and the sum and difference formulas for trigonometric functions remain valid. However, the absolute value of the complex cosine is not always less than or equal to 1, as is the case for the real cosine.
Through the complex exponential function, and possibly a calculator you can calculate function values. We give a few examples.
\[\begin{aligned}\sin(2.0-2.0\,\mathrm{i})&= \frac{e^{(2.0-2.0\,\mathrm{i})\,\mathrm{i}}-e^{-(2.0-2.0\,\mathrm{i})\,\mathrm{i}}}{2\mathrm{i}}\\ &\phantom{abcxyz}\color{blue}{\text{definition of trigonometric function}} \\ &= \frac{e^{2.0+2.0\,\mathrm{i}}-e^{-2.0-2.0\,\mathrm{i}}}{2\mathrm{i}} \\ &\phantom{abcxyz}\color{blue}{\text{complex arithmetic}}\\ &= \frac{e^{2.0}e^{2.0\,\mathrm{i}}-e^{-2.0}e^{-2.0\,\mathrm{i}}}{2\mathrm{i}}\\ &\phantom{abcxyz}\color{blue}{\text{simplification}}\\ &= \frac{e^{2.0}\bigl(\cos(2.0)+\sin(2.0)\mathrm{i}\bigr)-e^{-2.0}\bigl(\cos(-2.0)+\sin(-2.0)\mathrm{i}\bigr)}{2\mathrm{i}} \\ &\phantom{abcxyz}\color{blue}{\text{definition of exponentional function}}\\ &{\approx}\; 3.421+1.509\,\mathrm{i} \\ &\phantom{abcxyz}\color{blue}{\text{computing with a calculator}} \end{aligned}\]
