Precalculus / Unit 7: Polar Coordinates and Complex Numbers
7.2 Polar Equations and Rectangular Equations
Notes
- $r = k$ is a circle of radius $|k|$ centered at the origin. $\theta = \alpha$ is a line through the origin.
- Use $r^2 = x^2 + y^2$, $x = r\cos\theta$, $y = r\sin\theta$ to convert.
- $r = 6\cos\theta$: multiply by $r$ to get $r^2 = 6r\cos\theta$, so $x^2 + y^2 = 6x$, a circle centered at $(3, 0)$ with radius 3.
- Vertical line $x = 3$ becomes $r\cos\theta = 3$.
Practice
Which rectangular equation matches $r = 6\cos\theta$?
Multiply by $r$: $r^2 = 6r\cos\theta$.
Which polar equation describes the vertical line $x = 3$?
Substitute $x = r\cos\theta$.
Which polar equation describes the line $y = x$?
The line makes a $45^\circ$ angle with the x-axis, so $\displaystyle \theta = \frac{\pi}{4}$.
Ready for more? Take the Unit 7 test.