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

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

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

(4)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.