MAT1093 Lecture Notes: Polar Coordinates and Graphs

Polar Coordinates and Graphs

Introduction

  • Discussing different ways to graph: Rectangular vs. Polar coordinates.

Polar Graphs - Plotting

  • Plotting points in polar coordinates (r, θ).
    • Start at the center (origin).
    • Use the angle θ for direction.
    • Use the radius r for distance.
  • Example:
    • (r, θ) = (2, π/3)
  • Examples:
    • (5, π/2) - circle
    • (3, -π/2)
    • (-2, π/4)
  • How to plot (-2, π/4) with a positive radius r?
    • (2, 5π/4)

Converting Between Rectangular and Polar Coordinates

  • Rectangular and polar coordinates exist on the same plane.
  • Polar to Rectangular:
    • x=rcos⁡(θ)x = r \cos(\theta)
    • y=rsin⁡(θ)y = r \sin(\theta)
  • For a point (r, θ), use the above equations to get (x, y).
  • Example: Convert (3, π/2) to rectangular form.
    • x=3cos⁡(π2)=3∗0=0x = 3 \cos(\frac{\pi}{2}) = 3 * 0 = 0
    • y=3sin⁡(π2)=3∗1=3y = 3 \sin(\frac{\pi}{2}) = 3 * 1 = 3
    • (x, y) = (0, 3)

Rectangular to Polar

  • Find r:
    • r2=x2+y2  ⟹  r=x2+y2r^2 = x^2 + y^2 \implies r = \sqrt{x^2 + y^2}
  • Find θ:
    • tan⁡(θ)=yx  ⟹  θ=tan⁡−1(yx)\tan(\theta) = \frac{y}{x} \implies \theta = \tan^{-1}(\frac{y}{x})

Example: Convert (3, 3) to Polar Form

  1. What quadrant is (3, 3) in?
    • Quadrant I
  2. Find r:
    • r=32+32=9+9=18=32r = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
  3. Find θ:
    • θ=tan⁡−1(33)=tan⁡−1(1)=π4\theta = \tan^{-1}(\frac{3}{3}) = \tan^{-1}(1) = \frac{\pi}{4}
    • (r, θ) = (32,π4)(3\sqrt{2}, \frac{\pi}{4})

Converting Equations Between Polar and Rectangular Forms

  • Example: x2+y2=9x^2 + y^2 = 9 (circle)
    • r2=9r^2 = 9
    • r=3r = 3
  • Rectangular: x2+y2=9x^2 + y^2 = 9
  • Polar: r=3r = 3
  • Example: Convert y=5y = 5 to polar form.
    • y=rsin⁡(θ)y = r \sin(\theta)
    • rsin⁡(θ)=5r \sin(\theta) = 5
  • Example :
    • x=3x = 3

Graphing Polar Equations

  • Circles
    • r=3cos⁡(θ)r = 3 \cos(\theta)
    • r=−3cos⁡(θ)r = -3 \cos(\theta)
    • r=3sin⁡(θ)r = 3 \sin(\theta)
    • r=−3sin⁡(θ)r = -3 \sin(\theta)

Cardioids (Heart-shaped)

  • r=2+2cos⁡(θ)r = 2 + 2 \cos(\theta)
  • r=2−2cos⁡(θ)r = 2 - 2 \cos(\theta)
  • r=2+2sin⁡(θ)r = 2 + 2 \sin(\theta)
  • r=2−2sin⁡(θ)r = 2 - 2 \sin(\theta)

Limacons

  • With one inner loop
    • r=3+2cos⁡(θ)r = 3 + 2 \cos(\theta)
    • r=3−2cos⁡(θ)r = 3 - 2 \cos(\theta)
    • r=2+3cos⁡(θ)r = 2 + 3 \cos(\theta)
    • r=2−3cos⁡(θ)r = 2 - 3 \cos(\theta)
    • r=3+2sin⁡(θ)r = 3 + 2 \sin(\theta)
    • r=3−2sin⁡(θ)r = 3 - 2 \sin(\theta)
    • r=2+3sin⁡(θ)r = 2 + 3 \sin(\theta)
    • r=2−3sin⁡(θ)r = 2 - 3 \sin(\theta)

Lemniscates

  • r2=a2cos⁡(2θ)r^2 = a^2 \cos(2\theta)
  • r2=−a2cos⁡(2θ)r^2 = -a^2 \cos(2\theta)
  • r2=a2sin⁡(2θ)r^2 = a^2 \sin(2\theta)
  • r2=−a2sin⁡(2θ)r^2 = -a^2 \sin(2\theta)

Rose Curves

  • r=3cos⁡(4θ)r = 3 \cos(4\theta)
  • r=3sin⁡(5θ)r = 3 \sin(5\theta)
  • r=3sin⁡(4θ)r = 3 \sin(4\theta)
  • r=3cos⁡(5θ)r = 3 \cos(5\theta)

Reminders

  • Due 5/5 (Today):
    • Webwork 5.2
    • Module 13 Writing Assignment
  • Due 5/9 (Friday):
    • Webwork 5.3
    • Webwork 5.4
    • Module Itwriting Assignment
  • Due 5/11 (Sunday):
    • Exam #3 corrections
    • Final Exam Practice Problems
    • End of Semester Reflection
    • ALL ASSIGNMENTS CLOSE