Polar Coordinates, Multiple Representations & Polar–Rectangular Conversions

Polar Coordinates: Concept & Plane

• Polar coordinates: $(r,\theta)$ where
  • $r$ = directed distance from origin (can be negative ⇒ reflection across origin).
  • $\theta$ = angle measured from positive $x$–axis; positive = counter-clockwise, negative = clockwise.
• Motivation
  • Useful for describing points on circular/spherical surfaces (e.g.", “ball”/sphere surface).
• Polar plane visualization
  • Concentric circles represent equal radii.
  • Standard angles (unit-circle): $0,\;\pi/6,\;\pi/4,\;\pi/3,\;\pi/2,\;2\pi/3,\;3\pi/4,\;5\pi/6,\;\pi,\; \ldots ,2\pi$.
  • Quadrant identification identical to rectangular system.

Plotting Individual Polar Points (Examples)

• Conventions reiterated repeatedly: first number radius, second number angle.

1. $(3,\;7\pi/6)$

   • $7\pi/6$ in Quadrant III.
   • Move out 3 units ⇒ final point.

2. $(4,\;-4\pi/3)$

   • Clockwise four increments of $\pi/3$ → Quadrant IV.
   • Radius 4 ⇒ plot.

3. $(-5,\;-7\pi/4)$

   • Equivalent positive angle $=\pi/4$.
   • Negative radius ⇒ reflect point at $(5,\pi/4)$ through origin → Quadrant III.

4. $(-4,\;19\pi/3)$

   • Reduce angle: $19\pi/3-6\pi=\pi/3$ (subtract one full turn $=6\pi/3$).
   • Angle now $\pi/3$ (Quadrant I); negative radius ⇒ reflect to Quadrant III.

5. $(-1,\;-39\pi/4)$

   • One full turn $=8\pi/4$. $-39\pi/4+40\pi/4 = \pi/4$.
   • Equivalent angle $\pi/4$; radius –1 ⇒ reflect to Quadrant III.

Student Practice Points

• $(-3,\;-5\pi/3)$
  • Equivalent angle $=\pi/3$; negative radius ⇒ Quadrant II.
• $(-4,\;82\pi/3)$
  • $82\pi/3-78\pi/3=4\pi/3$ (subtract $6\pi$).
  • Negative radius ⇒ reflect $(4,4\pi/3)$ to Quadrant IV.

Multiple Representations of One Polar Point

Example A: Point originally $(4,\;\pi/3)$ (Quadrant I)

Some other valid names (many possible):

1. $(4,\;7\pi/3)$ (add $2\pi$)
2. $(4,\;-5\pi/3)$ (clockwise version)
3. $(-4,\;-2\pi/3)$ (use opposite radius & add $\pi$)
4. $(-4,\;4\pi/3)$ (positive angle variant of #3)
5. $(-4,\;-8\pi/3)$ (clockwise full turn then reflect)

Example B: Point originally $(-2,\;13\pi/4)$

• Reduced positive angle $=\pi/4$; reflect ⇒ Quadrant III.
  Some alternative names discovered:
• $(2,\;\pi/4)$
• $(-2,\;9\pi/4)$
• $(2,\;-7\pi/4)$
• $(-2,\;-3\pi/4)$, etc.

Practice Example C: $(3,\;-5\pi/6)$

• Point lives in Quadrant II.
• Sample equivalents students built:
  • $(3,\;7\pi/6)$ (add $2\pi$)
  • $(-3,\;\pi/6)$ (flip radius)
  • $(3,\;19\pi/6)$ (add $2\pi$ again)
  • $(-3,\;-11\pi/6)$ (flip & clockwise)

Key takeaway:
Add/Subtract integer multiples of $2\pi$ to $\theta$ OR flip the sign of $r$ with a $\pi$ shift to create unlimited names.

Converting Polar → Rectangular

Formulas (memorize):
$x = r\cos\theta \qquad y = r\sin\theta$

Example 1: $(6,\;\pi/6)$

• $x = 6\cos(\pi/6)=6\cdot(\sqrt3/2)=3\sqrt3$
• $y = 6\sin(\pi/6)=6\cdot(1/2)=3$
• Rectangular: $(3\sqrt3,\;3)$

Example 2: $(-3,\;-7\pi/4)$

• $\cos(-7\pi/4)=\sqrt2/2$, $\sin(-7\pi/4)=\sqrt2/2$
• $x=y=-3(\sqrt2/2)=-\dfrac{3\sqrt2}{2}$
• Rectangular: $(-\tfrac{3\sqrt2}{2},\;-\tfrac{3\sqrt2}{2})$

Example 3: $(-8,\;11\pi/3)$

• Reduce angle: $11\pi/3-2\pi=5\pi/3$.
• $\cos(5\pi/3)=1/2\,,\; \sin(5\pi/3)=-\sqrt3/2$
• $x=-8\cdot\tfrac12=-4$
• $y=-8\cdot(-\tfrac{\sqrt3}{2})=4\sqrt3$
• Rectangular: $(-4,\;4\sqrt3)$

Converting Rectangular → Polar (not detailed in lecture but implied)

• $r = \sqrt{x^2+y^2}$
• $\theta = \operatorname{atan2}(y,x)$ plus quadrant adjustment.

Techniques for Angle Reduction & Reflection

• Add/Subtract $2\pi$ (or $360^\circ$) until angle is in desired interval.
• To handle negative radius:
  • Replace $(r,\theta)$ with $(-r,\theta+\pi)$ to keep location.

Homework / Assessment Timeline (course-specific)

• Sections 4.1 → 5.1 homework due $\text{May }4$ (no extension).
• Quiz #4: $\text{May }4$.
• Test #3: $\text{May }6$ (review emailed by evening of recording day).
• Final Exam: $\text{May }11$.
• Bonus Opportunities for Final (submit evidence the day you finish final):
  1. Course Evaluation (website) – write ≥3 full sentences → $+3\%$.
  2. RateMyProfessor (2-3 sentences, choose correct color bar!) → $+2\%$.
     • Combined bonus up to $5\%$ on final; final score also replaces lowest test score.
• Administrative/ethical note: send screenshots May 11 only so emails aren’t lost.

Key Takeaways & Study Tips

• Always identify quadrant first; negative $r$ ⇒ reflect across origin.
• Keep $\theta$ within $[0,2\pi)$ for sanity but remember infinitely many representations exist.
• Convert efficiently by memorizing special-angle trig values.
• When simplifying $r\cos\theta$ or $r\sin\theta$, cancel factors only outside radicals.
• On exams professor will specify “find n different polar names”; any correct set accepted.