The Complex Cosine and Sine Functions

# The Complex Cosine and Sine Functions

We will now extend the real-valued sine and cosine functions to complex-valued functions. For reference, the graphs of the real-valued cosine (red) and sine (blue) functions are given below:

Recall from The Complex Exponential Function page that for any imaginary number $iy$ we have that:

(1)
\begin{align} \quad e^{iy} = \cos y + i \sin y \quad (*) \end{align}

So then since $\cos y$ is an even function and $\sin y$ is an odd function we have that:

(2)
\begin{align} \quad e^{- iy} &= \cos (-y) + i \sin (-y) \quad (**) \\ &= \cos y - i \sin y \end{align}

If we sum $(*)$ and $(**)$ we get:

(3)
\begin{align} \quad 2 \cos y & = e^{yi} + e^{-yi} \\ \quad \cos y & = \frac{e^{yi} + e^{-yi}}{2} \end{align}

And if we subtract $(*)$ and $(**)$ we get:

(4)
\begin{align} \quad 2i \sin y = e^{iy} - e^{-iy} \\ \quad \sin y = \frac{e^{iy} - e^{-iy}}{2i} \end{align}

With these two formulas identified, we can now define the complex cosine and sine functions.

 Definition: The Complex Cosine Function is defined for all $z \in \mathbb{C}$ by $\displaystyle{\cos z = \frac{e^{iz} + e^{-iz}}{2}}$ and the Complex Sine Function is defined for all $z \in \mathbb{C}$ by $\displaystyle{\sin z = \frac{e^{iz} - e^{-iz}}{2i}}$.

It is important to note that we have not actually verified that these formulas are sensible extensions to the real-valued cosine and sine functions. Indeed they are, and we will subsequently look at some of their properties.