Topic 6 of 8

Polar coordinates and complex numbers

Step 14 · Trigonometry and precalculus topics

Locate points with polar coordinates, convert between polar and rectangular form, and write complex numbers in polar form to multiply them and raise them to powers.

Polar coordinates

Polar coordinates (r, θ) locate a point by its distance r from the origin and its angle θ from the positive x-axis.

The polar point (4, π/3): 4 units from the origin at an angle of 60°. In x and y it is (2, 2√3).

Example: Placing a point

Describe where the point $\left(3, \frac{5\pi}{6}\right)$ is.

Turn 150° from the positive x-axis.
Go 3 units out in that direction.

AnswerIn quadrant II, 3 from the origin

Converting between polar and rectangular

Convert with the unit circle and the Pythagorean theorem.

x = r\cos\theta, \quad y = r\sin\theta \qquad r^{2} = x^{2} + y^{2}, \quad \tan\theta = \frac{y}{x}

Example: Polar to rectangular

Convert $\left(6, \frac{2\pi}{3}\right)$ to rectangular coordinates.

$x = 6\cos\frac{2\pi}{3} = 6 \cdot \left(-\frac{1}{2}\right) = -3$ .
$y = 6\sin\frac{2\pi}{3} = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3}$ .

Answer(−3, 3√3)

Example: Rectangular to polar

Convert (−2, 2) to polar coordinates.

r = √(4 + 4) = 2√2.
tan θ = −1, and the point is in quadrant II, so θ = $\frac{3\pi}{4}$ .

Answer $\left(2\sqrt{2}, \frac{3\pi}{4}\right)$

Example: An equation

Write r = 4 cos θ in rectangular form.

Multiply both sides by r: r² = 4r cos θ.
So x² + y² = 4x, which is (x − 2)² + y² = 4.

AnswerA circle with center (2, 0) and radius 2

Complex numbers in polar form

The complex number a + bi is the point (a, b). In polar form it is r(cos θ + i sin θ), where r is its distance from 0 and θ its angle.

Example: Polar form

Write 1 + i√3 in polar form.

r = √(1 + 3) = 2.
tan θ = √3, so θ = $\frac{\pi}{3}$ .

Answer $2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$

Multiplying complex numbers multiplies their lengths and adds their angles. Doing that again and again gives De Moivre's theorem.

\left[r(\cos\theta + i\sin\theta)\right]^{n} = r^{n}(\cos n\theta + i\sin n\theta)

Example: A power

Find (1 + i)⁸.

1 + i has r = √2 and θ = $\frac{\pi}{4}$ .
$(\sqrt{2})^{8} = 16$ , and $8 \times \frac{\pi}{4} = 2\pi$ .
16(cos 2π + i sin 2π) is 16.

Answer16

Try it yourself

Work each one on paper first, then open the answer to check.

Convert $\left(2, \frac{\pi}{2}\right)$ to rectangular coordinates.
Show answer
(0, 2)
Convert (5, π) to rectangular coordinates.
Show answer
(−5, 0)
Convert (3, 3) to polar coordinates.
Show answer
$\left(3\sqrt{2}, \frac{\pi}{4}\right)$
Convert (0, −4) to polar coordinates.
Show answer
$\left(4, \frac{3\pi}{2}\right)$
How far from 0 is 3 − 4i?
Show answer
5
Multiply 2(cos 20° + i sin 20°) by 3(cos 40° + i sin 40°).
Show answer
6(cos 60° + i sin 60°), which is 3 + 3√3 i