Polar and Cartesian conversion
Objective. Move between coordinate systems with quadrant awareness.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Polar coordinates describe a point by its distance r from the origin and angle θ from the positive x-axis, related to Cartesian coordinates by x = r cos θ and y = r sin θ, with r = √(x²+y²) and tan θ = y/x. Converting from Cartesian to polar requires checking the quadrant of (x,y) to select the correct θ, since the arctan function alone only gives angles in a limited range. For example, the point (−1,1) has r=√2, and lies in the second quadrant, so θ = 3π/4, not the raw arctan(−1) = −π/4.
Worked example
Convert the Cartesian point (−2, −2√3) into polar coordinates (r, θ), giving θ between 0 and 2π.
- 1r = √((−2)² + (−2√3)²) = √(4+12) = √16 = 4
- 2Reference angle: arctan(2√3/2) = arctan(√3) = π/3
- 3Both coordinates negative places the point in the third quadrant, so θ = π + π/3 = 4π/3
- 4Polar coordinates are (4, 4π/3)
Answer: (r, θ) = (4, 4π/3)
Concept mastery
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Practice
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Start practisingCommon mistake. Students use θ = arctan(y/x) directly without adjusting for quadrant, giving wrong angles for points with negative x. Always sketch or reason about the quadrant before stating the final θ.
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