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How to Graph Polar in Trig

The polar coordinate system is a method of defining points in space, much like the familiar Cartesian coordinate grid. Unlike the Cartesian plane, however, polar coordinates define a point by the distance it is from a fixed point and the angle formed between the point and an initial line segment. Where a Cartesian coordinate pair is set up in this manner: (x, y), a polar coordinate pair is set up like this: (r, θ), where r is the radial distance of the line segment from the origin point and θ is its angle above the initial segment. Polar graphs are circular or spiral in appearance.

Things You'll Need

Scientific calculator
Pencil
Graphing paper

Instructions

- 1
  Draw a T-chart with the variable of the equation, theta (θ), on the left side and the polar equation, (r), on the right side.
- 2
  Write several values for θ within the period of the equation. For example, if the polar equation is in terms of sine then the period will be 2PI, so values can include 0, PI/2, PI, 3PI/2 and 2PI.
- 3
  Substitute the predetermined values of theta into the polar equation and solve for r. For example, for the polar equation r = 5 - 5sin(θ) with theta values 0, PI/2, PI, 3PI/2 and 2PI, the values of r are found to be: r = 5, r = 0, r = 5, r = 10 and r = 5. This creates polar coordinates (r, θ) = (5, 0), (0, PI/2), (5, PI), (10, 3PI/2) and (5, 2PI).
- 4
  Plot the polar points onto a polar coordinate grid, in ascending θ order. For example, for a polar graph centered at the origin point (0, 0) with the x-axis as the initial segment, the point (5, 0) is located 5 horizontal units to the right, along the x-axis, and 0 vertical units above the x-axis. This is because r = 5, and the angle between r and the x-axis is 0 degrees.
- 5
  Connect each of the plotted points with a smooth, circular curve.

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