Precalculus / Unit 7: Polar Coordinates and Complex Numbers
7.1 Polar Coordinates and Conversions
Notes
A point in polar coordinates is $(r, \theta)$: distance $r$ from the origin and angle $\theta$ from the positive x-axis.
- Polar to rectangular: $x = r\cos\theta$, $y = r\sin\theta$.
- Rectangular to polar: $\displaystyle r = \sqrt{x^2 + y^2}$ and $\displaystyle \tan\theta = \frac{y}{x}$ (check the quadrant).
- A point has many polar names: $(r, \theta) = (r, \theta + 2\pi) = (-r, \theta + \pi)$.
- Example: $\displaystyle \left(4, \frac{\pi}{3}\right)$ gives $\displaystyle x = 4 \cdot \frac{1}{2} = 2$ and $\displaystyle y = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}$.
Practice
Convert the polar point $\displaystyle \left(4, \frac{\pi}{3}\right)$ to rectangular coordinates.
$\displaystyle x = 4\cos\frac{\pi}{3} = 2$ and $\displaystyle y = 4\sin\frac{\pi}{3} = 2\sqrt{3}$.
Convert $(-2, 2)$ to polar coordinates with $r > 0$ and $0 \le \theta < 2\pi$.
$\displaystyle r = \sqrt{4 + 4} = 2\sqrt{2}$, and the point is in quadrant II where $\tan\theta = -1$ gives $\displaystyle \frac{3\pi}{4}$.
Which polar point is the same as $\displaystyle \left(-3, \frac{\pi}{4}\right)$?
A negative $r$ points the opposite way, which adds $\pi$ to the angle.
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