1120_8 - Week 8 Outline Polar Coordinate System Examples of...

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Week 8OutlinePolar Coordinate SystemExamples of Polar CurvesTangents to Polar CurvesIntersections of Polar CurvesAreas Bounded by Polar CurvesArc Lengths for Polar CurvesPolar Coordinate SystemTo describe a point in the plane in polar coordinates, we give its distance from theorigin, calledr, and its angle in radians going counterclockwise from the positivex-axis,which we callθ. So, instead of using (x, y) in rectangular coordinates, we describe a pointby (r, θ).Remarks1. In polar coordinates, a point can have more than one description. The point (1,0) inrectangular coordinates has distance 1 from the origin and has angle 0 with the positivex-axis. So in polar coordinates, it is (r, θ) = (1,0). But it is also given by (1,2π), (1,4π),etc. Even wackier is the origin, which is given by (0,0.9π), (0,2.3π), (0,π170), etc. That isto say, (r, θ) can also be represented by (r, θ+ 2) and the origin by (0, θ) for anyθ.2.Sometimes we may extend the meaning of polar coordinates (r, θ) to the case inwhichris negative by agreeing that, the points (-r, θ) and (r, θ) lie on the same linethrough the origin and at the same distance|r|from the origin, but on opposite sides ofthe origin. In short, (-r, θ) represents the same point as (r, θ+π).Relationship between Rectangular and Polar CoordinatesBy Pythagorean theorem, we have the following basic equationsx=rcosθ, y=rsinθ, r=px2+y2.Also,tanθ=yx,assumingx6= 0.Remark1
These four equations can be easily read off the picture and so there is no need tomemorize them.Examples

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Calculus, Arc Length, Cartesian Coordinate System, Polar Coordinates, Coordinate system, Polar coordinate system, Conic section, dr d dr d

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