How to Convert From Rectangular Coordinates to Polar Coordinates (Precalculus - Trigonometry 38)

Overview

Professor Leonard explains how to convert rectangular coordinates (x, y) to polar coordinates (r, θ). He details the formulas r = √(x² + y²) and θ = tan⁻¹(y/x), emphasizing the critical need to adjust the angle θ based on the point's quadrant, as tan⁻¹ has a limited range (-π/2 to π/2). Adjustments involve adding π for quadrants II and III, and 2π for positive angles in quadrant IV.

Key takeaways

To convert rectangular (x, y) to polar (r, θ), use r = √(x² + y²) and θ = tan⁻¹(y/x).
The range of tan⁻¹ is limited to (-π/2, π/2), covering quadrants I and IV.
For points in quadrant II or III, add π to the angle obtained from tan⁻¹.
For points in quadrant IV that require a positive angle, add 2π to the negative angle from tan⁻¹.
When dealing with decimal or non-unit circle values, use a calculator for θ and adjust based on the plotted quadrant.
Always plot the point first to determine the correct quadrant and necessary angle adjustments.

Chapters

0:00 Introduction to Rectangular to Polar Conversion

Previous video covered polar to rectangular conversion.
This video focuses on rectangular to polar conversion.
Ambiguity exists in rectangular to polar conversion, unlike polar to rectangular.

1:43 Formulas for Polar Coordinate Conversion

Rectangular to polar uses r = √(x² + y²) and θ = tan⁻¹(y/x).
r represents the distance from the origin (radius).
θ represents the angle from the positive x-axis.

3:33 Understanding tan⁻¹ Limitations and Quadrant Adjustments

tan⁻¹ is defined only for angles between -π/2 and π/2 (quadrants I and IV).
Points in quadrants I and IV are handled directly by tan⁻¹.
Points in quadrants II and III require adding π to the tan⁻¹ result.
Points in quadrant IV may yield negative angles; add 2π for a positive angle.

7:01 Example 1: Converting (3, 0) to Polar Coordinates

Identify x=3, y=0.
Calculate r = √(3² + 0²) = 3.
Calculate θ = tan⁻¹(0/3) = tan⁻¹(0) = 0.
Point (3, 0) is on the positive x-axis, θ=0 is correct.

15:38 Example 2: Converting (0, 2) to Polar Coordinates

Identify x=0, y=2.
Calculate r = √(0² + 2²) = 2.
tan⁻¹(2/0) is undefined; consider angles where tangent is undefined (π/2, 3π/2).
Plotting (0, 2) shows it's on the positive y-axis, so θ = π/2 is correct.

28:24 Example 3: Converting (1, -1) to Polar Coordinates

Identify x=1, y=-1.
Plotting shows the point is in quadrant IV.
Calculate r = √(1² + (-1)²) = √2.
Calculate θ = tan⁻¹(-1/1) = tan⁻¹(-1) = -π/4.
Since the point is in quadrant IV, -π/4 is the correct angle.
For a positive angle, add 2π: -π/4 + 2π = 7π/4.

41:45 Example 4: Converting (-3, 3) to Polar Coordinates

Identify x=-3, y=3.
Plotting shows the point is in quadrant II.
Calculate r = √((-3)² + 3²) = √18 = 3√2.
Calculate θ = tan⁻¹(3/-3) = tan⁻¹(-1) = -π/4.
Since the point is in quadrant II, add π to -π/4: -π/4 + π = 3π/4.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Leonard.