Converting Between Polar and Rectangular Coordinates

Written by tutor Barbara W.

What are rectangular and polar coordinates? They may both refer to the same locations (or points) on a chart or graph. I liken rectangular coordinates,
sometimes called Cartesian coordinates, to city walking directions and polar coordinates to airplane flying directions (“as the crow flies”)…

How did I figure out that the length of the line (0,0) to (4,4) is approximately 5.66 blocks? I used the Pythagorean Theorem. In the example below, we know
the triangle created with a and b of equal length (in this case 4 blocks) is known to be a 45° right triangle and remember
that a2 + b2 = c2, so 42 + 42 = 32, so c = √32 (or 4√2) which is approximately 5.66.

It’s easy to convert rectangular coordinates to polar coordinates when the angle of the polar coordinate is 0°, 30°, 45°, 60°, or 90°. These will
all be positive X,Y rectangular coordinates in Quadrant I of the Cartesian plane (X headed right from 0 and Y headed up from 0). Why? Because we can determine the
degree portion of the polar coordinates through our algebraic knowledge of special right triangles.

But we can only find the distance portion of the polar coordinates (through the Pythagorean theorem) when it is not a special right triangle! To find out the
degree ?? of the angle (see the Example 2), we now need trigonometry.

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