Cartesian to polar coordinates

If we have a Cartesian coordinate in the form (x,y), then we can convert these to a polar coordinate, using our [[ /polar coordinate identities|polar coordinate identities]] from before!

General steps

polar coordinates are in the form (r,\theta), where:

• r is the magnitude, the distance of a point from the origin
• \theta is the bearing from the origin, starting from the right-stretching line and going anticlockwise.

To find the magnitude of the polar coordinate, r:

• Just use the Pythagorean theorem to find the hypotenuse!
• r=\sqrt{x^2+y^2}

Then to find the angle, \theta:

• \theta=\tan^{-1}(\frac yx)

Note: If the x coordinate is negative, we need to add \pi to our answer for \theta. That’s because, otherwise, the angle we got from \tan^{-1}(\frac yx) would be in the wrong quadrant.

Find the polar coordinate from the Cartesian (4,4)

• r=\sqrt{4^2+4^2}
• r=\pm\sqrt32
• r=\pm4\sqrt2
• \theta=\tan^{-1}(\frac44)
• \theta=\tan^{-1}(1)
• \theta=\frac{\pi}4
• Answer: (4\sqrt2, \frac{\pi}4)

Convert the cartesian coordinate (-1,4) to polar coordinates (to 3sf)

• r=\sqrt{(-1)^2+4^2}
• r=\sqrt{17}
• r\approx4.123
• \theta=\tan^{-1}(\frac4{-1})
• \theta=\tan^{-1}4{-1}
• \theta=\tan^{-1}(4)
• \theta=-\tan^{-1}(-4)
• \theta=-1.326
• Because our x coordinate is negative, we need to add \pi:
  • \theta=-1.326+\pi
  • \theta\approx1.816
• Answer: (4.123, 1.816)

Convert the cartesian coordinate (-1,-4) to polar coordinates (to 3sf)

• r=\sqrt{(-1)^2+(-4)^2}
• r=\sqrt{17}
• r\approx4.123
• \theta=\tan^{-1}(\frac{-4}{-1})
• \theta=\tan^{-1}4
• \theta=-\tan^{-1}(-4)
• \theta=-1.326
• Because our x coordinate is negative, we need to add \pi
• \theta=-1.326+\pi
• \theta\approx1.816
• Answer: (4.123, 1.816)

Convert the cartesian coordinate (1,-4) to polar coordinates (to 3sf)

• r=\sqrt{1^2+(-4)^2}
• r=\sqrt{17}
• r\approx4.123
• \theta=\tan^{-1}(\frac{-4}{1})
• \theta=\tan^{-1}(-4)
• \theta=-\tan^{-1}4
• \theta=-1.326
• Because our x coordinate is positive, we don’t need to add \pi!
• Answer: (4.123, -1.326)

Write (4,8) as a set of polar coordinates

• r=\sqrt{4^2+8^2}
• r=\sqrt{80}
• r=4\sqrt5
• \theta=\tan^{-1}(\frac8{4})
• \theta=\tan^{-1}2
• \theta=1.107
• Answer: (4\sqrt5, 1.107)