AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.13 — Trigonometry and Polar Coordinates
Notes — A New Way to Locate Points
💡 Learning Objectives (3.13.A)
By the end of this lesson you will be able to:
- Plot points (r, θ) on a polar coordinate grid
- Convert between polar and rectangular (Cartesian) coordinates
- Recognize that a single point has many polar representations
- Convert simple equations between rectangular and polar form
1. Polar Coordinates Defined
In RECTANGULAR coordinates, we locate a point using horizontal x and vertical y. In POLAR coordinates, we locate the same point using:
- r — the (signed) distance from the origin (the ‘pole’)
- θ — the angle measured from the positive x-axis (the ‘polar axis’) to the ray through the point
A polar coordinate is written (r, θ).
2. Converting Polar → Rectangular
If a point has polar coordinates (r, θ), then its rectangular coordinates are:
x = r cos θ, y = r sin θ
📘 Example — Polar to rectangular
- (4, π/3): x = 4 cos(π/3) = 4(1/2) = 2; y = 4 sin(π/3) = 4(√3/2) = 2√3. Rectangular (2, 2√3)
- (3, π): x = 3 cos π = −3; y = 3 sin π = 0. Rectangular (−3, 0)
- (−2, π/2): x = −2 cos(π/2) = 0; y = −2 sin(π/2) = −2. Rectangular (0, −2)
3. Converting Rectangular → Polar
Going the other way is slightly trickier because there are infinitely many polar representations of each point. The basic formulas:
r = ±√(x² + y²), tan θ = y/x (be careful with the quadrant!)
📘 Example — Rectangular to polar
- Convert (1, 1). r = √(1 + 1) = √2. tan θ = 1/1 = 1, and the point is in Quadrant I, so θ = π/4. Polar: (√2, π/4)
- Convert (−1, √3). r = √(1 + 3) = 2. tan θ = √3 / (−1) = −√3. The point is in Quadrant II, so θ = π − π/3 = 2π/3. Polar: (2, 2π/3)
4. Multiple Representations
Unlike rectangular coordinates (one (x, y) per point), polar coordinates are NOT unique. Three things produce the same point:
- Adding 2π to θ: (r, θ) ≡ (r, θ + 2π)
- Negating r and adding π: (r, θ) ≡ (−r, θ + π)
- Both at once: (r, θ) ≡ (−r, θ − π) ≡ (r, θ + 4π) etc.
📘 Example — Many names for one point
(2, π/3) is the same point as:
- (2, π/3 + 2π) = (2, 7π/3)
- (−2, π/3 + π) = (−2, 4π/3)
- (2, π/3 − 2π) = (2, −5π/3)
📝 What does negative r mean?
If r < 0, you face direction θ but then walk BACKWARD by |r| units. So (−3, π/4) is the same point as (3, π/4 + π) = (3, 5π/4).
5. The Pole
The origin (0, 0) in rectangular coordinates is called the POLE in polar coordinates, and has many representations: (0, θ) is the pole for EVERY value of θ. This is because the distance is zero, so the angle is irrelevant.
6. Converting Equations
Sometimes we want to rewrite an equation in polar form (using r and θ instead of x and y) or vice versa. Use:
- x = r cos θ
- y = r sin θ
- x² + y² = r²
- y/x = tan θ
📘 Example — Rectangular ↔ Polar equations
- Circle x² + y² = 9: substitute x² + y² = r². Polar form: r² = 9, or r = 3
- Line y = 2x: rewrite as y/x = 2, so tan θ = 2, i.e., θ = arctan 2
- Polar r = 4 cos θ: multiply by r: r² = 4r cos θ, i.e., x² + y² = 4x. Rewrite: (x − 2)² + y² = 4 (a circle of radius 2 centered at (2, 0))
7. Summary
- Polar coordinate (r, θ): r is signed distance from pole, θ is angle from positive x-axis
- Conversion: x = r cos θ, y = r sin θ; r² = x² + y², tan θ = y/x
- Many polar representations correspond to the same point — be careful when comparing
- Use substitution to convert between polar and rectangular equations
- Negative r reverses direction; the pole (0, θ) represents the origin for every θ