# Converting between Polar and Rectangular Coordinates

Definition: Rectangular and polar coordinates are both ways of defining the location of a given point.

Rectangular coordinates, in the form (x, y), indicate a point’s horizontal (x) and vertical (y) distance from the origin.

Polar coordinates, in the form (rθ), indicate a point’s radial distance (r) from the pole (origin), and the measure of the angle (θ) formed counter-clockwise from the polar axis (positive x-axis) to the point.

To convert polar coordinates to rectangular coordinates,

The point (r, θ) will become (r cosθ, r sinθ):

$\dpi{150} \bg_white (rcos\theta , rsin\theta ) \rightarrow (x,y)$

To convert rectangular coordinates to polar coordinates,

The point (x, y) will become (√(x²+y²), tan¯¹(y/x) ):

$\dpi{150} \bg_white (\sqrt{x^2+y^2},\;tan^{-1}\;\frac{y}{x})\;\rightarrow\; (r,\theta)$

Example 1: Convert (2, (3π/4) ) to rectangular coordinates.

Solution:

Step 1: Plug (2, (3π/4) ) into the formula and simplify, where r = 2 and θ = 3π/4:

$\dpi{150} \bg_white \\(rcos\theta, rsin\theta)\\ \\(2\;cos\frac{3\pi}{4}, \;2\;sin\frac{3\pi}{4})\\ \\= (2\cdot -\frac{\sqrt2}{2}\;,\;2\cdot \frac{\sqrt2}{2}\;)\\ \\= (-\sqrt2, \;\sqrt2)$

Example 2: Convert (2, -3) into polar coordinates.

Solution:

Step 1:  Plug (2, -3) into the formula and simplify:

$\dpi{150} \bg_white \\(\sqrt{x^2+y^2},\;tan^{-1}\;\frac{y}{x})\\ \\(\sqrt{2^2+(-3)^2},\;tan^{-1}\;\frac{\!\!-3}{\;2})\\ \\=(\sqrt{13},\ {-.98})$

The value of θ, -.98, can also be written as a positive radian value by adding 2π, to get (√13, 5.3).

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