Lesson 13.2

Polar to Rectangular Conversion

Two coordinate systems, one point. Learn to translate between polar and rectangular forms—a skill essential for calculus and physics.

Introduction

Now that you can plot polar points, the next step is converting between and . The unit circle provides the key relationships.

1

Prerequisite Connection

You can plot polar points as .

2

Today's Increment

We learn the conversion formulas between polar and rectangular coordinate systems.

3

Why This Matters

Calculus uses both systems. You'll integrate in polar when shapes are circular, but need rectangular for most algebra.

Conversion Formulas

Polar → Rectangular

Rectangular → Polar

⚠️ Finding θ Correctly

Using only gives the correct angle in Quadrants I and IV. For Quadrants II and III, add to the result. Always check which quadrant your point is in!

Useful Identities for Equations

Worked Examples

Example 1: Converting a Point to Rectangular

Convert to rectangular coordinates.

Step 1: Apply x = r cos θ

Step 2: Apply y = r sin θ

Solution

Example 2: Converting a Point to Polar

Convert to polar coordinates with and .

Step 1: Find r

Step 2: Find reference angle

Step 3: Determine the quadrant

The point is in Quadrant III. So .

Solution

Example 3: Converting an Equation

Convert the polar equation to rectangular form.

Step 1: Multiply both sides by r

Step 2: Substitute identities

Use and :

Step 3: Complete the square

Solution

This is a circle centered at with radius 2.

Common Pitfalls

Using arctan without checking quadrant

returns values in . You must add for points in Quadrants II or III.

Forgetting to multiply by r before substituting

When converting equations, multiplying by creates terms like that substitute cleanly.

Taking only positive square root for r

While we often want , remember that is also valid and represents the same point with a different angle.

Real-World Application

Navigation and GPS

GPS satellites transmit position in latitude/longitude (a spherical coordinate system), but your phone's map shows rectangular coordinates (a flat projection).

The conversion between these systems uses similar formulas. Engineers must constantly translate between coordinate systems to display accurate maps and calculate routes.

Practice Quiz

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