Polar Coordinates Math Example 2

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

Example 2

medium
Convert the rectangular point (โˆ’3,3)(-3, 3) to polar coordinates with r>0r > 0 and 0โ‰คฮธ<2ฯ€0 \le \theta < 2\pi.

Solution

  1. 1
    Find rr: r=x2+y2=9+9=32r = \sqrt{x^2 + y^2} = \sqrt{9 + 9} = 3\sqrt{2}.
  2. 2
    Find the reference angle: tanโกโˆ’1(โˆฃyโˆฃโˆฃxโˆฃ)=tanโกโˆ’1(1)=ฯ€4\tan^{-1}\left(\frac{|y|}{|x|}\right) = \tan^{-1}(1) = \frac{\pi}{4}.
  3. 3
    The point (โˆ’3,3)(-3, 3) is in Quadrant II, so ฮธ=ฯ€โˆ’ฯ€4=3ฯ€4\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}.

Answer

(32,3ฯ€4)\left(3\sqrt{2}, \frac{3\pi}{4}\right)
Converting from rectangular to polar requires finding both rr (using the Pythagorean theorem) and ฮธ\theta (using arctangent and quadrant analysis). The quadrant determines the correct angle since tanโกโˆ’1\tan^{-1} only returns values in (โˆ’ฯ€2,ฯ€2)(-\frac{\pi}{2}, \frac{\pi}{2}).

About Polar Coordinates

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

Learn more about Polar Coordinates โ†’

More Polar Coordinates Examples

Example 1 easy

Convert the polar coordinates [formula] to rectangular (Cartesian) coordinates.

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].

โ† All Polar Coordinates Examples Polar Coordinates Concept