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4.1: Polar Coordinates

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Identify major parts of the polar coordinate system: the pole, polar axis, θ, r
  • Plot points in the polar coordinate system
  • Use a graphing utility to plot common polar functions

Pole, Polar Axis, θ, r

Symmetry can be explored by using trigonometric functions and a coordinate system called the Polar Coordinate System.

The Polar Coordinate System consists of a ray known as the Polar Axis and the endpoint of the ray, called the Pole.

The coordinate (r, θ), where r is a distance from the Pole and θ is the angle a line segment from the pole to the point makes with the Polar Axis. The point (r, θ) is known as the Polar Coordinate.

Plot (θ, r)

To plot a point, first go along the r-axis by r units. Then, rotate counterclockwise by the angle θ. Be careful to use the correct units for the angle measure (either radians or degrees).

Examples of polar coordinates are point A \begin{align*}\left (2,\frac{\pi}{3}\right )\end{align*}, point B (4, 135o), and point C \begin{align*}\left (-2,\frac{\pi}{6}\right )\end{align*}.

Below is the pole, polar axis and the points A, B and C.

Usually polar plots are done with radians (especially if they include trigonometric functions), but sometimes degrees are used.

If r is negative you can plot a point (r, θ) by first plotting (| r |, θ), and then rotating by π = 180o.

Sinusoids of One Revolution (e.g., limaçons, cardioids)

As with the rectangular coordinate system, equations can be graphed in the polar coordinate system. We will go into more detail on graphing in a future section. For now, these graphs are examples of what shapes are possible to graph as polar functions.

Below is the graph of a polar equation r = 1 + 2 sin θ (limaçon)

Graph: r = 1 + cos θ (cardioid)

Applications, Technological Tools

Polar Equations can be graphed using a graphing calculator: With the graphing calculator- go to MODE. There select RADIAN for the angle measure and POL (for Polar) on the FUNC (function)line. When Y = is pressed, note that the equation has changed from y = to r = . There input the polar equation. After pressing graph, if you can’t see the full graph, adjust x- and y- max/min, etc in WINDOW.

Lesson Summary

Until now, most, if not all, graphing has been done using the rectangular coordinate system. There are other coordinate systems, such as polar coordinates, with which graphing can be accomplished.

Points to Consider

Some equations are easier to graph in one coordinate system as opposed to the other coordinate system. Knowing how to graph in each coordinate system allow the freedom to be able to graph any equation.

Review Questions

  1. Plot the following points: a. (4, 30o) b. (2.5, π) c. \begin{align*}\left(-1,\frac{\pi}{3}\right )\end{align*} d. \begin{align*}\left(3,\frac{5\pi}{6}\right )\end{align*} e. (-2, 300o)
  2. Use a graphing calculator or plotting program to plot the following equations: a. 1 + 3 sin θ b. 1 + 2 cos θ

Review Answers

  1. a. b.

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