Converting Between Polar and Rectangular Forms

Introduction

In trigonometry, it is often necessary to convert equations and coordinates between polar and rectangular (Cartesian) forms. This process is essential for analyzing curves and solving equations in different coordinate systems.

Key Formulas for Conversion

• Rectangular to Polar:

• Polar to Rectangular:

  • Given and , use the above formulas to find and .

Example 1: Converting Equations

Convert the equation to polar form:

• Recall:

• So,

• Therefore,

Interpretation: This is the equation of a circle of radius 2 centered at the origin in both rectangular and polar forms.

Example 2: Converting from Polar to Rectangular

• Given:

• Multiply both sides by :

• Recall: and

• So:

• Rewriting:

• Complete the square:

Interpretation: This is a circle of radius 1 centered at in rectangular coordinates.

Example 3: Converting from Rectangular to Polar

• Given:

• Recall: ,

• Substitute:

• Simplify:

• Since , substitute:

• Thus, the equation is satisfied for all where .

Summary Table: Conversion Formulas

Additional info:

• These conversions are fundamental in trigonometry, especially for graphing and analyzing curves such as circles, lines, and conic sections in different coordinate systems.

• Always consider the domain and range of when interpreting polar equations.