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Polar-Rectangular Conversion Formulas

Polar-Rectangular Conversion Formulas

Rules for converting between polar coordinates and rectangular coordinates.

 

Diagram showing polar-rectangular conversion formulas: x=rcosθ, y=rsinθ, r²=x²+y², tanθ=y/x, with labeled right triangle...

Key Formula

x=rcos⁡θ,y=rsin⁡θr=x2+y2,θ=tan⁡−1 ⁣(yx)x = r\cos\theta, \quad y = r\sin\theta \qquad r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\!\left(\frac{y}{x}\right)
Where:
  • xx = Horizontal coordinate in the rectangular system
  • yy = Vertical coordinate in the rectangular system
  • rr = Distance from the origin to the point (radius)
  • θ\theta = Angle measured counterclockwise from the positive x-axis

Worked Example

Problem: Convert the polar point (r, θ) = (6, 60°) to rectangular coordinates.
Step 1: Find the x-coordinate using the cosine formula.
x=rcos⁡θ=6cos⁡60°=6⋅12=3x = r\cos\theta = 6\cos 60° = 6 \cdot \frac{1}{2} = 3
Step 2: Find the y-coordinate using the sine formula.
y=rsin⁡θ=6sin⁡60°=6⋅32=33y = r\sin\theta = 6\sin 60° = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3}
Answer: The rectangular coordinates are (3,  33)(3,\; 3\sqrt{3}), which is approximately (3,  5.196)(3,\; 5.196).

Why It Matters

Many curves, such as spirals and cardioids, have much simpler equations in polar form, so converting between systems lets you choose whichever representation makes a problem easier. These formulas also appear when you evaluate double integrals by switching to polar coordinates, a standard technique in calculus for regions with circular symmetry.

Common Mistakes

Mistake: Using θ = tan⁻¹(y/x) without checking the quadrant of the point.
Correction: The inverse tangent function returns values only in (−90°, 90°). If the point lies in quadrant II or III, you must add 180° (or π) to get the correct angle. Always plot or check the signs of x and y to determine the true quadrant.

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