Instead of using $x$ and $y$ coordinates, polar coordinates describe a point by its distance $r$ from the origin and the angle $\theta$ measured counterclockwise from the positive $x$-axis. This system is especially useful for curves that have circular or spiral symmetry.
🎯 In this section you will learn
The relationship between polar and rectangular coordinates
How to convert points and equations between the two systems
How to sketch basic polar curves: circles, cardioids, roses, limaçons
📌 Polar vs. Rectangular Coordinates
$$ x = r\cos\theta \quad \text{and} \quad y = r\sin\theta $$
$$ r^2 = x^2 + y^2 \quad \text{and} \quad \tan\theta = \frac{y}{x} $$
💡 Important Notes
$r$ can be negative — this means the point is in the opposite direction of $\theta$.
$\theta$ is typically measured in radians.
Polar coordinates are not unique: $(r, \theta)$ and $(r, \theta + 2\pi)$ represent the same point.
📊 Visualizing Polar Curves
Interactive graph below. See how different polar equations create different shapes:
$r = 2$ (Circle)
$r = 1 + \cos\theta$ (Cardioid)
$r = \cos(2\theta)$ (Rose)
$r = 2 + \cos\theta$ (Limaçon)
Convert $(r, \theta) = (4, \pi/3)$ to rectangular coordinates.
$x = r\cos\theta = 4\cos(\pi/3) = 4 \cdot \frac{1}{2} = 2$
$y = r\sin\theta = 4\sin(\pi/3) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}$
Point: $(2, 2\sqrt{3})$
Convert $(x, y) = (-3, 3)$ to polar coordinates.
$r = \sqrt{x^2 + y^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$
$\tan\theta = \frac{y}{x} = \frac{3}{-3} = -1$ → $\theta = \frac{3\pi}{4}$ (since $x<0, y>0$, Quadrant II)
Point: $(3\sqrt{2}, 3\pi/4)$
Convert $r = 4\sin\theta$ to rectangular form.
Multiply both sides by $r$: $r^2 = 4r\sin\theta$
$x^2 + y^2 = 4y$ → $x^2 + y^2 - 4y = 0$ → $x^2 + (y - 2)^2 = 4$
This is a circle centered at $(0, 2)$ with radius $2$.
Sketch $r = 2\cos\theta$.
Multiply by $r$: $r^2 = 2r\cos\theta$ → $x^2 + y^2 = 2x$ → $(x - 1)^2 + y^2 = 1$
This is a circle centered at $(1, 0)$ with radius $1$.
⚠️ Common Mistakes
Negative $r$: Remember that $(-r, \theta)$ represents the same point as $(r, \theta + \pi)$.Incorrect quadrant when finding $\theta$: Always check the signs of $x$ and $y$ to determine the correct quadrant.Multiplying polar equations: When converting to rectangular, sometimes multiplying both sides by $r$ is helpful.
🔍 Key Takeaways
Polar coordinates: $(r, \theta)$ where $r$ is distance from origin, $\theta$ is angle.
Conversion: $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$, $\tan\theta = y/x$.
Common polar curves: circles, cardioids ($r = a \pm a\cos\theta$), roses ($r = a\cos(n\theta)$), limaçons ($r = a + b\cos\theta$).
← Back to Module Page