DWG NO. 6.6 — Lesson 6 of 6

Polar Coordinates and Polar Graphs

Unit 6: Additional Topics in Trigonometry · ~25 min

Every point you've plotted so far has used rectangular coordinates (x, y) — two perpendicular distances from the origin. Polar coordinates locate a point a different way: by a distance r from a fixed point called the pole, and an angle θ measured from a fixed ray called the polar axis. For curves built around rotation or distance from a center, polar equations are often dramatically simpler than their rectangular equivalents.

Converting between systems

Overlay the pole on the origin and the polar axis on the positive x-axis, and the two systems connect through the same right-triangle relationships as the unit circle:

• x = r cosθ   and   y = r sinθ   (polar → rectangular)
• r² = x² + y²   and   tanθ = \(\frac{y}{x}\)   (rectangular → polar, check the quadrant of (x, y) to place θ)

Common polar graphs

Guided practice

• 1.Convert (r, θ) = (8, 210°) to rectangular coordinates.
  Show answer

  x = 8cos210° ≈ −6.93, y = 8sin210° = −4. (−6.93, −4).
• 2.Convert (0, −5) to polar coordinates.
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  r = 5, and the point sits straight down the negative y-axis, so θ = 270°.
• 3.Convert (−3, 3) to polar coordinates with 0° ≤ θ < 360°.
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  r = \(\sqrt{9+9}\) = 3\(\sqrt{2}\) ≈ 4.24. Reference angle 45°; Quadrant II gives θ = 135°.
• 4.Identify the type of curve given by r = 3(1 + cosθ).
  Show answer

  This matches the form a(1 + cosθ), so it's a cardioid.
• 5.Explain why a single point can be written with infinitely many different polar coordinate pairs.
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  Adding any multiple of 360° to θ sweeps back to the same ray, so (r, θ) and (r, θ + 360°n) describe the same point for any integer n. A negative r also reaches the same point as a positive r rotated 180°, giving (−r, θ + 180°) as another equivalent pair — so there's no single "correct" representation the way there is in rectangular coordinates.