Section 6.4: Intro to the Polar Plane

The Basics—Plotting Points

• All points in the polar coordinate system are

centered around the origin—the pole, point O,
and rotate from the polar axis. Each point P is
(r, θ), where r is the directed distance and θ is
the angle of rotation.

Find All Polar Coordinates of A Point

• If the point P has polar coordinates (3, π/3),

find all the polar coordinates for P.

Formulas:
(r, θ + 2nπ) or (-r, θ + (2n + 1)π)

​All co-terminal points work:
(3, π/3 + 2πn)

But you must also think about the reflection:
(-3, π/3 + π + 2πn) = (-3, 4π/3 + 2πn)

Coordinate Conversions

Being able to convert between polar and rectangular coordinates is super helpful and pretty straight-forward.

Given any point, we can use our prior trig. knowledge to convert its form with the following equations:
x = r cos θ
y = r sin θ
r² = x² + y²
Tan θ = y/x

Converting Equations

Convert the given polar equation to rectangular form: r = 4 cos θ

STEPS:
1. Multiply both sides by r to create r²: r² = (4 cos θ)r
2. Reorder the RHS to allow for substitution: r² = 4 (r cos θ)
3. Substitute x² + y² in for r² and x for r cosθ: x² + y² = 4x
4. Solve for y² or y for final answer:

ANSWER: y² = -x² + 4x or y = +√(-x² + 4x)

Convert From Rectangular to Polar: …

• (x – 3)² + (y – 2)² = 13

• Step 1: Expand—

• Step 2: Combine Like Terms—

• Step 3: Substitute—

• Step 4: Factor—

• FINAL ANSWER: