Section 6.4: Intro to the Polar Plane

Section 6.4:
Polar Coordinates
and 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.

Multiple Choice

What is the value of point A?

• (7π/6, 4)

• (-4, 7π/6)

• (4, 7π/6)

• (7π/6, -4)

Multiple Choice

What is the value of point E?

• (4π/3, -6)

• (6, π/3)

• (6, 4π/3)

• (π/3, -6)

Multiple Select

What is the value of point C? (Check all that apply)

• (7, 5π/3)

• (7, π/3)

• (-7, 2π/3)

• (-7, 8π/3)

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

Multiple Choice

Find the polar coordinates of the rectangular point (0,3)

• (3, π)

• (3, 0)

• (3, π/2)

• (3, -3π/2)

Multiple Choice

Find the polar coordinates of the rectangular point (√3, 1)

• (2, π/6)

• (-2, -π/6)

• (1, 0)

• (4π, -2)

Multiple Choice

Find the polar coordinates of the rectangular point: (6√3, 6)

• (12, π/6)

• (6, 7π/6)

• (6, π/6)

• (12, 7π/6)

Multiple Choice

Find the rectangular coordinates of a polar point: (6, 2π/3)

• (-3, 3)

• (3, 3√3)

• (-3, 3√3)

• (3, -3√3)

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: