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Multivariable Calculus · Question of the Day

Multivariable Calculus Question of the Day

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Saturday, September 5, 2026

In cylindrical coordinates, the equation r=2sin⁡(θ)r = 2\sin(\theta) represents a curve. What is the Cartesian equation of this same curve?

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Question of the Day

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In cylindrical coordinates, the equation r=2sin⁡(θ)r = 2\sin(\theta) represents a curve. What is the Cartesian equation of this same curve?

  1. x2+y2=2yx^2 + y^2 = 2y
  2. (x−1)2+y2=1(x-1)^2 + y^2 = 1
  3. x2+(y−1)2=1x^2 + (y-1)^2 = 1 (correct answer)
  4. x2+y2=2xx^2 + y^2 = 2x

Explanation: Starting with r=2sin⁡(θ)r = 2\sin(\theta), multiply both sides by rr: r2=2rsin⁡(θ)r^2 = 2r\sin(\theta). Since r2=x2+y2r^2 = x^2 + y^2 and rsin⁡(θ)=yr\sin(\theta) = y, we get x2+y2=2yx^2 + y^2 = 2y. Rearranging: x2+y2−2y=0x^2 + y^2 - 2y = 0. Completing the square: x2+(y2−2y+1)=1x^2 + (y^2 - 2y + 1) = 1, which gives x2+(y−1)2=1x^2 + (y-1)^2 = 1. This is a circle with center (0,1)(0,1) and radius 11. Option A shows the intermediate step before completing the square. Option B represents a circle centered at (1,0)(1,0), which would come from r=2cos⁡(θ)r = 2\cos(\theta). Option D would result from r=2cos⁡(θ)r = 2\cos(\theta) before completing the square.