Overview

Exponential form converts trig products into sums and reveals hidden symmetries.

Core Skills

Switch Between Trig and Exponential

Use $e^{i\theta}$ to rewrite trig expressions into algebraic ones.

Use Conjugates

For real-valued expressions, pair $e^{i\theta}$ with $e^{-i\theta}$ to cancel imaginary parts.

Recognize $z+1/z$ Patterns

If $z=e^{i\theta}$, then $z+1/z=2\cos\theta$ and $z-1/z=2i\sin\theta$.

Key Ideas

• $\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$.
• $\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$.
• Expressions like $z + 1/z$ collapse to $2\cos\theta$ when $z=e^{i\theta}$.

Worked Example

If $z + \frac{1}{z} = 2\cos 20^\circ$, compute $z^{18} + z^{-18}$.

Let $z = e^{i\cdot 20^\circ}$. Then $z^{18} + z^{-18} = e^{i\cdot 360^\circ} + e^{-i\cdot 360^\circ} = 1 + 1 = 2$.

More Examples

Example 1: Cosine as Exponential

Compute $\cos 3\theta$ in terms of $e^{i\theta}$.

$\cos 3\theta = \frac{e^{i3\theta}+e^{-i3\theta}}{2}$.

Example 2: Sum of Powers

If $z=e^{i\theta}$, simplify $z^2+z^{-2}$.

$2\cos 2\theta$.

Example 3: Real Part

Find $\Re\left(e^{i\theta}(1+i)\right)$.

$\Re\left((\cos\theta+i\sin\theta)(1+i)\right)=\cos\theta-\sin\theta$.

Strategy Checklist

• Convert to $e^{i\theta}$ form early.
• Pair conjugates to get real values.
• Reduce angles modulo $2\pi$.

Common Pitfalls

• Mixing degree and radian measures.
• Forgetting that $z$ may be $e^{-i\theta}$ as well.
• Dropping the factor of $2$ when converting to cosine.

Practice Problems