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Polar Coordinates

Polar coordinates are (r, \theta) where r is the directed distance from the pole to the point and and \theta is the directed angle from the polar axis to \overrightarrow{OP}

To convert rectangular coordinates to polar coordinates given (x, y)

r = \sqrt{x^{2}+y^{2}}

\theta = tan^{-1}(\frac{y}{x}) \quad when x > 0

\theta = tan^{-1}(\frac{y}{x} ) \quad when x < 0

or

\theta = tan^{-1}(\frac{y}{x})+180^{\circ}

Example:

Convert (7, 10) to polar coordinates

To convert to polar coordinates, first we will write down our values for x and y so we can plug it into the formula later. x = 7 and y = 10

r = \sqrt{x^{2}}{y^{2}} = \sqrt{(7)^{2}+(10)^{2}} = \sqrt{49+100} = \sqrt{149} = 12.21

\theta = tan^{-1}(\frac{y}{x}) = 0.96

so our answer in polar coordinates is: (12.21, 0.96)