Polar Coordinates Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Convert the polar coordinates (4,π3)(4, \frac{\pi}{3}) to rectangular (Cartesian) coordinates.

Solution

  1. 1
    Use the conversion formulas: x=rcos⁡θx = r\cos\theta and y=rsin⁡θy = r\sin\theta.
  2. 2
    x=4cos⁡(π3)=4⋅12=2x = 4\cos\left(\frac{\pi}{3}\right) = 4 \cdot \frac{1}{2} = 2.
  3. 3
    y=4sin⁡(π3)=4⋅32=23y = 4\sin\left(\frac{\pi}{3}\right) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}.

Answer

(2,23)(2, 2\sqrt{3})
Polar coordinates (r,θ)(r, \theta) locate a point by its distance from the origin and angle from the positive xx-axis. Converting to rectangular uses x=rcos⁡θx = r\cos\theta and y=rsin⁡θy = r\sin\theta, which come from the right triangle formed by the point, the origin, and the projection onto the xx-axis.

About Polar Coordinates

A coordinate system where each point in the plane is described by a distance rr from the origin and an angle θ\theta from the positive xx-axis, written as (r,θ)(r, \theta).

Learn more about Polar Coordinates →

More Polar Coordinates Examples

Example 2 medium

Convert the rectangular point [formula] to polar coordinates with [formula] and [formula].

Example 3 medium

Convert the equation [formula] to polar form.

Example 4 hard

Find all polar representations of the point with rectangular coordinates [formula] where [formula].

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