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Unformatted text preview: Section 8.1
Polar Coordinates Polar Coordinate System
In the rectangular coordinate system we represent points by an ordered pair of numbers, (x,y) where x represents the (signed) distance from the yaxis and y the distance from the xaxis. To create our axis's we select an origin and create two perpendicular lines through it. Polar Coordinate System
In the polar coordinate system we represent points by an ordered pair of numbers, (r,) where r represents the (signed) distance from a point we call the pole, and represents the angle measure from a ray called the polar axis. To create our axis we select a pole, i.e. an origin, and create a ray starting at the pole. Examples
(1,) (2,) (1/2,) (2,) (1,/2) (2,2) (1/2,) (2,/4) (1,/2) (2,2) (1/2,) (2,/4) (0,) (0,/16) Notice that with any point in polar coordinates (r,) can also be represented by (r, + 2k) and(r, + + 2k) Also the polar coordinates of the pole are (0,) for any . Theorem: Conversion from Polar to Rectangular
If P is a point with polar coordinates (r,), the rectangular coordinates (x,y) of P, assuming the pole lies on the origin, are given by x = r cos y = r sin The proof of this is straightforward since we have defined sin = y/r and cos = x/r, we just have to consider when r is negative. Proof
Since (r,) = (r, + ) we can use the second set to calculate what the x and y values are sin( + ) = sin() = y/r cos( + ) = cos() = x/r Thus we get the correct coordinates. Converting Rectangular to Polar Coordinates
Step 1: Plot the point (x,y), or at the bare minimum note what quadrant the point is in. Step 2: Use the formula r2 = x2 + y2 to find r. Step 3: For quadrant 1 use = tan1y/x For quadrant 2 use = + tan1y/x For quadrant 3 use = + tan1y/x For quadrant 4 use = tan1y/x ...
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