Precalculus / Unit 7: Polar Coordinates and Complex Numbers
7.5 Polar Form of Complex Numbers
Notes
A complex number can be written $z = r(\cos\theta + i\sin\theta)$, often shortened to $r\,\text{cis}\,\theta$, where $r = |z|$ and $\theta$ is the argument.
- Example: $\displaystyle 1 + i = \sqrt{2}\,\text{cis}\,\frac{\pi}{4}$.
- Multiplying: multiply the moduli and add the angles.
- Dividing: divide the moduli and subtract the angles.
- Example: $\displaystyle 3\,\text{cis}\,\frac{\pi}{6} \cdot 2\,\text{cis}\,\frac{\pi}{3} = 6\,\text{cis}\,\frac{\pi}{2} = 6i$.
Practice
What is the polar form of $1 + i$?
$\displaystyle r = \sqrt{1 + 1} = \sqrt{2}$ and $\tan\theta = 1$ in quadrant I.
Write $\displaystyle 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ in the form $a + bi$.
$\displaystyle 2 \cdot \frac{1}{2} = 1$ and $\displaystyle 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}$.
What is $\displaystyle 3\,\text{cis}\,\frac{\pi}{6} \cdot 2\,\text{cis}\,\frac{\pi}{3}$?
Multiply moduli: 6. Add angles: $\displaystyle \frac{\pi}{2}$, and $\displaystyle \text{cis}\,\frac{\pi}{2} = i$.
Ready for more? Take the Unit 7 test.