9.1.1: Resources and Key Concepts

Prerequisite Topics

This page lists the prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Quadratic Functions
  • The Basics of Quadratic Functions and Vertex Form: This skill is required to convert some polar equations into the standard form of a circle in rectangular coordinates, as seen in Example 8.
• Graphing Inequalities
  • Graphing Linear Inequalities in Two Variables: The concept of graphing regions in the plane defined by inequalities is extended from the Cartesian system to the polar system in this section (Example 4).

Videos

• Polar Coordinates
  • Polar Coordinates - The Polar Coordinate System
• Converting Coordinates
  • Polar Coordinates - Converting Ordered Pairs from Polar to Rectangular Coordinates
  • Polar Coordinates - Converting Ordered Pairs from Rectangular to Polar Coordinates
• Polar Equations
  • Polar Coordinates - Converting Equations from Polar to Rectangular Form
  • Polar Coordinates - Converting Equations from Rectangular to Polar Form

Definitions

• Polar Coordinate System: A two-dimensional system for specifying points in a plane using a distance from a central point (the pole) and an angle from a reference direction (the polar axis).
• Pole: The central point in the polar coordinate system, equivalent to the origin in the rectangular coordinate system.
• Polar Axis: The reference direction from the pole, equivalent to the positive x-axis in the rectangular coordinate system.
• Polar Coordinates: The ordered pair \((r, \theta)\) that describes the location of a point, where \(|r|\) is the distance from the point to the pole and \(\theta\) is the signed angle from the polar axis to the ray passing through the point.

Theorems

• Cartesian Coordinates in Terms of Cosine and Sine: If a point \(P\) is located at a distance \(r\) from the origin in the direction specified by an angle \(\theta\) in standard position, then the Cartesian coordinates of \(P\) are \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
• Non-Uniqueness of Polar Coordinates: Any point with polar coordinates \((r, \theta)\) can also be represented by \((r, \theta + 2k\pi)\) for any integer \(k\), or by \((-r, \theta + \pi)\). The pole has coordinates \((0, \theta)\) for any angle \(\theta\).

Common Mistakes

• Polar Backwardness: It is customary to write polar coordinates as \((r, \theta)\), which lists the dependent variable \(r\) before the independent variable \(\theta\). This is the reverse of the standard \((x, y)\) convention for rectangular coordinates.
• Reference Angles and Quadrants: When converting from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), the angle \(\theta\) is not simply \(\arctan(\frac{y}{x})\). The arctangent function only returns angles in Quadrants I or IV. You must use the quadrant of the point \((x, y)\) to find the correct angle \(\theta\).
• Plotting Negative r-values: A negative value for \(r\) in polar coordinates, such as \((-2, \frac{\pi}{4})\), means you move in the direction opposite to the angle. In this case, you would face the direction \(\frac{\pi}{4}\) but move 2 units in the opposite direction, ending up on the ray for \(\frac{5\pi}{4}\).