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 (r, θ), 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):
To convert rectangular coordinates to polar coordinates,
The point (x, y) will become (√(x²+y²), tan¯¹(y/x) ):
Example 1: Convert (2, (3π/4) ) to rectangular coordinates.
Step 1: Plug (2, (3π/4) ) into the formula and simplify, where r = 2 and θ = 3π/4:
Example 2: Convert (2, -3) into polar coordinates.
Step 1: Plug (2, -3) into the formula and simplify:
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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