Step by Step
1
What polar coordinates represent
Instead of (x,y), a polar coordinate (r,θ) describes a point using r (distance from the origin) and θ (angle measured from the positive x-axis).
2
Converting polar to rectangular
x = r·cos(θ), and y = r·sin(θ) — this directly follows from the unit circle definition of cosine and sine, scaled by r.
3
Converting rectangular to polar
r = √(x²+y²), and tan(θ) = y/x — but you must check which quadrant (x,y) is actually in before finalizing θ, since the inverse tangent function alone doesn't distinguish between certain quadrant pairs.
4
Polar coordinates are not unique
Unlike rectangular coordinates, a single point has infinitely many valid polar representations. For example, (r,θ) and (−r, θ+π) describe the exact same physical point, since a negative r means going the opposite direction of θ, and adding π (180°) also reverses direction.
Applied Walkthrough
1
Convert the polar coordinate (4, 60°) to rectangular coordinates. Apply x = r·cos(θ) and y = r·sin(θ).
2
x = 4·cos(60°) = 4·(1/2) = 2. y = 4·sin(60°) = 4·(√3/2) = 2√3.
3
So the rectangular coordinates are (2, 2√3).
4
Note this same point could also be described in polar form as (−4, 240°), since a negative radius combined with an angle 180° away from the original points in the exact same direction — illustrating why polar representations aren't unique.
Exam Application
Exams test whether you can correctly convert between polar and rectangular coordinates in both directions, and whether you understand that polar representations of a single point are not unique.
⚠ Common Trap
The most common trap is computing θ using arctan(y/x) without checking which quadrant the point actually lies in — arctan alone can't distinguish between angles that differ by 180°, so the quadrant must be checked separately to get the correct angle.
✓ Quick Self-Check
1. How do you convert polar coordinates (r,θ) to rectangular coordinates?
x = r·cos(θ), y = r·sin(θ).
Tap to reveal / hide
2. How do you find r when converting rectangular coordinates to polar?
r = √(x²+y²).
Tap to reveal / hide
3. Why must you check the quadrant when finding θ using arctan(y/x)?
Because arctan alone can't distinguish between angles that differ by 180° — the quadrant of (x,y) must be checked to determine the correct angle.
Tap to reveal / hide
4. Convert (4, 60°) from polar to rectangular coordinates.
(2, 2√3).
Tap to reveal / hide
5. Why are polar coordinate representations of a point not unique?
Because a negative r combined with an angle shifted by π describes the same point as a positive r with the original angle.
Tap to reveal / hide