Introduction
You've spent years plotting points as . Now we're adding a new tool: polar coordinates, written as .
Prerequisite Connection
You know the unit circle and can evaluate trig functions at any angle.
Today's Increment
We learn to plot points using distance from the origin and angle from the positive x-axis.
Why This Matters
Polar coordinates are essential for physics (circular motion), engineering (radar systems), and calculus (double integrals in polar form).
Polar Coordinates
Polar Coordinate Form
= distance from origin (pole), = angle from positive x-axis (polar axis)
When r is Positive
The point lies at distance in the direction of angle .
When r is Negative
Move units in the opposite direction of .
Multiple Representations
Unlike Cartesian coordinates, polar points have infinitely many representations:
where is any integer
Interactive Polar Coordinate System
Adjust and to see how polar coordinates locate points. The concentric circles show distance from the origin, and the radial lines show angles.
Polar
(3, 0.25π)
Rectangular
(2.12, 2.12)
Worked Examples
Example 1: Plotting a Point with Positive r
Plot the point .
Step 1: Identify r and θ
(positive),
Step 2: Rotate from positive x-axis
Rotate counterclockwise by from the positive x-axis.
Step 3: Move outward
From the origin, move 3 units in the direction of .
Result
The point is in Quadrant I, 3 units from the origin at a 45° angle.
Example 2: Plotting with Negative r
Plot the point .
Step 1: Identify r and θ
(negative!),
Step 2: Find the reference direction
The angle points into Quadrant I.
Step 3: Move in the opposite direction
Since is negative, move 2 units in the opposite direction (into Quadrant III).
Result
Equivalent to
Example 3: Multiple Representations
Find three different polar representations for the point .
Representation 1: Add 2π
Representation 2: Subtract 2π
Representation 3: Use negative r
All Equivalent
, , , and all represent the same point.
Common Pitfalls
Forgetting that negative r exists
Unlike distances in geometry, can be negative in polar coordinates. This reflects the point through the origin.
Assuming unique representation
Each polar point has infinitely many names. When checking if two points are equal, convert both to the same form.
Using degrees when radians are expected
Most calculus applications use radians. Always check which unit is required.
Real-World Application
Radar and Sonar Systems
Radar systems use polar coordinates naturally. The antenna rotates (measuring angle ) and sends out pulses that return after hitting objects (measuring distance ).
Air traffic controllers see aircraft as polar coordinates: "Flight 247 is at 12 nautical miles, bearing 045°"—which translates directly to .
Practice Quiz
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