Properties of the Complex Cosine and Sine Functions

Properties of the Complex Cosine and Sine Functions

Recall from The Complex Cosine and Sine Functions page that if $z \in \mathbb{C}$ then the complex cosine and complex sine functions are defined respectively as:

(1)
\begin{align} \quad \cos z = \frac{e^{iz} + e^{-iz}}{2} \quad \quad , \quad \quad \sin z = \frac{e^{iz} - e^{-iz}}{2i} \end{align}

Notice that if $z \in \mathbb{R}$, i.e., $z = x + 0i$, then the complex cosine function reduces to the real cosine function:

(2)
\begin{align} \quad \cos z = \frac{e^{ix} + e^{-ix}}{2} = \frac{1}{2} \left [ \left ( \cos x + i \sin x \right ) + \left ( \cos (-x) + i \sin (-x) \right ) \right ] = \frac{1}{2} \left [ 2 \cos x \right ] = \cos x \end{align}

Similarly, the complex sine function reduces to the real sine function:

(3)
\begin{align} \quad \sin z = \frac{e^{ix} - e^{-ix}}{2i} = \frac{1}{2i} \left [ \left ( \cos x + i \sin x \right ) - \left ( \cos (-x) + i \sin (-x) \right ) \right ] = \frac{1}{2i} \left [ 2i \sin x \right ] = \sin x \end{align}

We will now look at some properties of the complex cosine and sine functions. It is important to note that while these properties may be obvious for the real-valued cosine and sine functions - they are not obvious for the complex-valued cosine and sine functions until we prove them.

Proposition 1 (The Pythagorean Identity): If $z \in \mathbb{C}$ then $\sin^2 z + \cos^2 z = 1$.
  • Proof: We use the formulas for $\cos z$ and $\sin z$ directly to get:
(4)
\begin{align} \quad \sin^2 z + \cos^2 z &= \left ( \frac{e^{iz} - e^{-iz}}{2i} \right )^2 + \left ( \frac{e^{iz} + e^{-iz}}{2} \right )^2 \\ &= \frac{(e^{iz} - e^{-iz})^2}{-4} + \frac{(e^{iy} + e^{-iz})^2}{4} \\ &= -\frac{e^{iz}e^{iz} -2e{iz}e^{-iz} + e^{-iz}e^{-iz}}{4} + \frac{e^{iz}e^{iz} +2e^{iz}e^{-iz} + e^{-iz}e^{-iz}}{4} \\ &=\frac{4e^{iz}e^{-iz}}{4} \\ &= e^{iz}e^{-iz} \\ &= e^{iz - iz} \\ &= e^{i(z - z)} \\ &= e^{0} \\ &= 1 \end{align}
Proposition 2: If $z, w \in \mathbb{C}$ then:
a) $\sin (z + w) = \sin z \cos w + \cos z \sin w$.
b) $\cos (z + w) = \cos z \cos w - \sin z \sin w$.
  • Proof of a) Let $z, w \in \mathbb{C}$. Then:
(5)
\begin{align} \quad \sin z \cos w + \cos z \sin w &= \left ( \frac{e^{iz} - e^{-iz}}{2i} \cdot \frac{e^{iw} + e^{-iw}}{2}\right ) + \left ( \frac{e^{iz} + e^{-iz}}{2} \cdot \frac{e^{iw} - e^{-iw}}{2i} \right ) \\ &= \frac{e^{iz}e^{iw} + e^{iz}e^{-iw} - e^{-iz}e^{iw} - e^{-iz}e^{-iw}}{4i} + \frac{e^{iz}e^{iw} - e^{iz}e^{-iw} + e^{-iz}e^{iw} - e^{-iz}e^{-iw}}{4i} \\ &= \frac{e^{i(z + w)} - e^{-i(z + w)}}{2i} + \frac{e^{i(z + w)} - e^{-i(z + w)}}{4i} \\ &= \frac{2e^{i(z + w)} - 2e^{-i(z + w)}}{4i} \\ &= \frac{e^{i(z + w)} - e^{-i(z + w)}}{2i} \\ &= \sin (z + w) \quad \blacksquare \end{align}
  • Proof of b) Analogous to that of (a).

The following proposition mimics the "odd" property of the real-valued sine function and the "even" property of the real-valued cosine function.

Proposition 3: For all $z \in \mathbb{C}$:
a) $\sin (-z) = -\sin (z)$.
b) $\cos (-z) = \cos (z)$.
  • Proof of a) We have that:
(6)
\begin{align} \quad \sin (-z) = \frac{e^{i(-z)} - e^{-i(-z)}}{2i} = \frac{e^{-iz} - e^{iz}}{2i} = -\frac{e^{iz} - e^{-iz}}{2i} = -\sin (z) \end{align}
  • Proof of b) We have that:
(7)
\begin{align} \quad \cos (-z) = \frac{e^{i(-z)} + e^{-i(-z)}}{2} = \frac{e^{-iz} + e^{iz}}{2} = \frac{e^{iz} + e^{-iz}}{2} = \cos (z) \end{align}
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