1/15
3.13 · Polar vs Rectangular
Compare polar (r,θ) and rectangular (x,y) coordinate systems. What does each coordinate represent in polar?
r = ? · θ = ?
2/15
3.13 · Plotting Polar Points
How do you plot A(3, 2π/3) on a polar grid? Describe step by step.
r=3 means which ring? θ=2π/3 means which direction?
3/15
3.13 · Negative r
What does a negative r mean in polar coordinates? Where does (−2, π/4) land?
Travel in the OPPOSITE direction of θ
4/15
3.13 · Equivalent Reps
List all three families of equivalent polar representations for point (r, θ).
Add 2πk · add π and negate r · subtract π and negate r
5/15
3.13 · Example 2 — Check
Verify that (2, −7π/4) represents the same point as (2, π/4).
Add 2π to the angle
6/15
3.13 · Example 3
Write two equivalent representations for (3, 5π/6).
One with same r, one with negative r
7/15
3.13 · Polar→Rect Formulas
State the formulas to convert polar (r,θ) to rectangular (x,y).
From cosθ=x/r and sinθ=y/r
8/15
3.13 · Rect→Polar Formulas
State the formulas to convert rectangular (x,y) to polar (r,θ).
Pythagorean theorem + arctan — but watch the quadrant!
2/15
✓ Plotting Polar Points
A(3, 2π/3): go to ring r=3, face angle 2π/3=120° (Q2). Mark the intersection.
r tells you WHICH ring (circle). θ tells you WHICH direction. Go to ring 3, rotate 120° counterclockwise from the polar axis.
1/15
✓ Polar vs Rectangular
r = distance from origin (radius). θ = angle from polar axis (direction). Both together uniquely specify the point.
Rectangular uses (x,y): go right x, up y (road grid). Polar uses (r,θ): face direction θ, travel distance r (flying directly).
4/15
✓ Equivalent Representations
(r, θ+2πk) · (−r, θ+π) · (−r, θ−π)
Adding 2π = same direction, full circle. Adding π + negating r = travel backwards in the flipped direction. Both land on the same physical point.
3/15
✓ Negative r
(−2, π/4) travels in the opposite direction of 45°, landing in Q3 at the same distance as (2, 5π/4).
Negative r: flip direction 180°. (−r, θ) = (r, θ+π). So (−2, π/4) = (2, π/4+π) = (2, 5π/4). Same point, same distance from origin.
6/15
✓ Example 3 — Two Equivalents
Any two of: (3,−7π/6) · (−3,−π/6) · (−3,11π/6)
From (3, 5π/6): same r: 5π/6−2π=−7π/6. Neg r, −π: 5π/6−π=−π/6, r=−3. Neg r, +π: 5π/6+π=11π/6, r=−3.
5/15
✓ Example 2 — Verify
−7π/4 + 2π = −7π/4 + 8π/4 = π/4 ✓. Both (2, π/4) and (2, −7π/4) land at the same point.
Adding 2π to any angle gives the same terminal direction — you've completed one full revolution and returned to start.
8/15
✓ Rect→Polar Formulas
r = √(x²+y²) · tanθ = y/x (then use sketch to find correct quadrant!)
arctan(y/x) only gives values in (−π/2, π/2) — Q1/Q4. If the point is in Q2 or Q3, add π to the arctan result. ALWAYS sketch first.
7/15
✓ Polar→Rect Formulas
x = r cosθ · y = r sinθ
Derived from: cosθ=x/r → x=rcosθ. sinθ=y/r → y=rsinθ. Just plug in r and θ, then evaluate the trig functions.
9/15
3.13 · Convert P→R
Convert (4, 5π/3) from polar to rectangular.
x=rcosθ, y=rsinθ
10/15
3.13 · Convert R→P
Convert (−2, 2) from rectangular to polar.
Sketch first! Q2 point.
11/15
3.13 · Convert R→P axis pt
Convert (0, −5) from rectangular to polar.
Special case — on the negative y-axis
12/15
3.13 · Negative r Example
Convert (−3, 7π/6) from polar to rectangular.
Apply x=rcosθ, y=rsinθ with r=−3
13/15
3.13 · Complex: Rectangular
How is the complex number a+bi represented as a point? As a polar form?
Point (a,b) · polar: rcosθ + i(rsinθ)
14/15
3.13 · Complex: Polar→Rect
Express 4cos(7π/6)+i·4sin(7π/6) in rectangular form.
r=4, θ=7π/6. Use x=rcosθ, y=rsinθ.
15/15
3.13 · Complex: Rect→Polar
Express the complex number −2−2i in polar form.
Find r=√(a²+b²) and θ. Point (−2,−2) is in Q3.
10/15
✓ Convert (−2,2) Rect→Polar
(2√2, 3π/4)
r=√(4+4)=√8=2√2. tanθ=2/(−2)=−1. Q2 point → θ=π−π/4=3π/4. (Not −π/4 which is Q4!)
9/15
✓ Convert (4, 5π/3) Polar→Rect
(2, −2√3)
x=4cos(5π/3)=4(1/2)=2. y=4sin(5π/3)=4(−√3/2)=−2√3.
12/15
✓ Convert (−3, 7π/6) Polar→Rect
(3√3/2, 3/2)
x=−3cos(7π/6)=−3(−√3/2)=3√3/2. y=−3sin(7π/6)=−3(−1/2)=3/2. The negative r flips both signs.
11/15
✓ Convert (0,−5) Rect→Polar
(5, 3π/2)
r=√(0+25)=5. Point is on the negative y-axis → θ=3π/2 (pointing straight down). tanθ=−5/0 is undefined — special axis case.
14/15
✓ Complex Polar→Rect
(x,y) = (−2√3, −2) · Complex: −2√3 − 2i
x=4cos(7π/6)=4(−√3/2)=−2√3. y=4sin(7π/6)=4(−1/2)=−2.
13/15
✓ Complex: Representations
Rectangular: a+bi ↔ point (a,b). Polar: r cosθ + i(r sinθ) where r=√(a²+b²).
3+2i ↔ point (3,2). To convert rect→polar: r=√(9+4)=√13, θ=arctan(2/3) (Q1, so no adjustment needed).
15/15
✓ Complex Rect→Polar (−2−2i)
r=2√2, θ=5π/4. Polar form: √8·cos(5π/4)+i·(√8·sin(5π/4))
r=√(4+4)=√8=2√2. Point (−2,−2) in Q3: tanθ=1, arctan(1)=π/4, add π for Q3: θ=5π/4.