Defining Polar Coordinates

Trigonometry · Axiom Academy

A new coordinate system where points are located using distance and angle instead of horizontal and vertical positions. In Cartesian coordinates , we describe a point using how far right/left (x) and up/down (y) it is from the origin. In polar coordinates , we instead describe how far away the point is (r) and what angle (θ) it makes. 2. The Radius (r): Distance from Origin The first coordinate in polar form is r , the radius or distance from the origin to the point. It's always measured as a straight line from (0, 0) to your point. r = 0 means the point is at the origin Larger r means farther from the origin r is typically non-negative (r ≥ 0) 3. The Angle (θ): Direction from Positive x-axis The second coordinate is θ (theta), the angle measured counterclockwise from the positive x-axis to the line connecting the origin to the point. Measured counterclockwise from the positive x-axis θ = 0° points to the right (positive x-axis) θ = 90° points up (positive y-axis) 4. Plotting a Point in Polar Coordinates To plot a point given in polar form (r, θ), follow these steps: Rotate by angle θ from the positive x-axis Move distance r along that direction Here's something interesting: The same point can have multiple polar coordinate representations! This is because you can add or subtract full rotations (360° or 2π) to the angle.

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