Polar Coordinates

Polar Coordinates - Polar Coordinates Up to this point weve...

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Polar Coordinates Up to this point we’ve dealt exclusively with the Cartesian (or Rectangular, or x-y ) coordinate system. However, as we will see, this is not always the easiest coordinate system to work in. So, in this section we will start looking at the polar coordinate system. Coordinate systems are really nothing more than a way to define a point in space. For instance in the Cartesian coordinate system at point is given the coordinates ( x,y ) and we use this to define the point by starting at the origin and then moving x units horizontally followed by y units vertically. This is shown in the sketch below. This is not, however, the only way to define a point in two dimensional space. Instead of moving vertically and horizontally from the origin to get to the point we could instead go straight out of the origin until we hit the point and then determine the angle this line makes with the positive x -axis. We could then use the distance of the point from the origin and the amount we needed to rotate from the positive x -axis as the coordinates of the point. This is shown in the sketch below. Coordinates in this form are called polar coordinates .
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The above discussion may lead one to think that r must be a positive number. However, we also allow r to be negative. Below is a sketch of the two points and . From this sketch we can see that if r is positive the point will be in the same quadrant as θ. On the other hand if r is negative the point will end up in the quadrant exactly opposite θ. Notice as well that the coordinates describe the same point as the coordinates do. The coordinates tells us to rotate an angle of from the positive x -axis, this would put us on the dashed line in the sketch above, and then move out a distance of 2. This leads to an important difference between Cartesian coordinates and polar coordinates. In Cartesian coordinates there is exactly one set of coordinates for any given point. With polar coordinates this isn’t true. In polar coordinates there is literally an infinite number of coordinates for a given point. For instance, the following four points are all coordinates for the same point.
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Here is a sketch of the angles used in these four sets of coordinates. In the second coordinate pair we rotated in a clock-wise direction to get to the point. We shouldn’t forget about rotating in the clock-wise direction. Sometimes it’s what
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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.

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Polar Coordinates - Polar Coordinates Up to this point weve...

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