CalcRef

Parametric, Polar & Vectors · Polar Coordinates

Polar to Cartesian

x=rcos⁡θ,y=rsin⁡θ,r2=x2+y2,tan⁡θ=y/xx = r\cos\theta, \quad y = r\sin\theta, \quad r^2 = x^2 + y^2, \quad \tan\theta = y/x

Conversion formulas between polar coordinates (r, θ) and Cartesian coordinates (x, y).

Worked examples

Convert (r, θ) = (4, π/3) to Cartesian.
  1. x = 4 cos(π/3) = 4(1/2) = 2
  2. y = 4 sin(π/3) = 4(√3/2) = 2√3

Answer: (2, 2√3)

Convert (x, y) = (-1, 1) to polar.
  1. r = √(1+1) = √2
  2. θ = arctan(1/(-1)) = arctan(-1). Since point is in Q2, θ = 3π/4.

Answer: (√2, 3π/4)

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