Rucete ✏ AP Precalculus In a Nutshell
9. Polar Functions
This chapter introduces the polar coordinate system and how to graph polar equations. You’ll learn to convert between polar and rectangular coordinates, analyze polar graphs, explore polar forms of complex numbers, and describe changes in radius with respect to angle.
- Polar Coordinates and Definitions
- Polar system uses points (r, θ), where r is the distance from origin (pole), and θ is the angle from the polar axis (positive x-axis).
- Positive θ → counterclockwise; Negative θ → clockwise.
- Positive r → point lies on terminal side; Negative r → opposite direction.
- Same point can have multiple polar representations.
- Coordinate Conversion
- From polar to rectangular:
- x = r cos θ
- y = r sin θ
- From rectangular to polar:
- r = √(x² + y²)
- θ = tan⁻¹(y/x), adjusted by quadrant
- Examples:
- (r, θ) = (5, π/2) → (x, y) = (0, 5)
- (x, y) = (−5, −12) → (r, θ) = (13, 4.318)
- Complex Numbers in Polar Form
- Complex number a + bi can be written as r(cos θ + i sin θ)
- r = √(a² + b²), θ = argument of complex number
- Examples:
- z = 1 − i → polar form = √2(cos 7π/4 + i sin 7π/4)
- z = −4i → polar form = 4(cos 3π/2 + i sin 3π/2)
- Graphing Polar Functions
- Graph r = f(θ) by plotting radius r at angle θ
- Examples:
- r = 3 → circle of radius 3
- r = θ → spiral
- r = 6 cos θ → circle across quadrants
- r = 2 + 4 cos θ → limaçon
- Use θr-plane to sketch and analyze
- Interpreting Polar Graph Behavior
- As θ increases, r changes to form curve shape
- Positive increasing r → distance grows
- Positive decreasing r → distance shrinks
- Negative r values → reflect direction
- Graph cardioids, rose curves, limaçons by segments
- Minimum = closest to origin; Maximum = farthest away
- Average Rate of Change in Polar Functions
- Formula: Δr / Δθ over an interval
- Shows how fast radius changes per radian
- Example:
- r = 4 − 2 sin θ
- θ from 0 to π/2: r drops from 4 to 2
- θ from π/2 to π: r rises from 2 to 4
- Analyze each interval to interpret shape
In a Nutshell
Polar functions offer a new way to describe curves using angles and radii. You can graph spirals, circles, and limaçons using polar equations. Conversion between rectangular and polar forms, especially with complex numbers, deepens your understanding of geometry and algebra. Mastering polar graphs and their behavior helps you model rotational and radial systems with precision.