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30 - x 2 y 2-4 x = 0 1.1 Curve Sketching Example 1.4 Trace...

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Lecture 3 January 10, 2012 1 Polar coordinates Occasionally, it is useful to represent a curve in a different set of coordinates than cartesian coordinates. One common coordinate system is polar coordinates. For this system, we represent a point in the plane, R 2 , by an angle, θ and a radius r . ADD PICTURE Thus, x = r cos θ and y = r sin θ. We can use the above two equations to define r and θ given x and y . Usually 0 θ < 2 π and r 0. To find r and θ given x and y we use r = p x 2 + y 2 and θ = tan - 1 x y . Example 1.1. The curve x 2 + y 2 = 1 is given by r = 1 in polar coordinates. ADD PICTURE Example 1.2. Transform the equation r 2 = a 2 cos(2 θ ) to cartesian coordinates. Example 1.3. Find the polar equation of the curve which is described by

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Unformatted text preview: x 2 + y 2-4 x = 0 1.1 Curve Sketching Example 1.4. Trace the curve r = 1 + cos θ . Note that we only need to consider θ ∈ [0 , 2 π ) . Why? Example 1.5. Trace the curve r 2 = sin(2 θ ) . 1 Example 1.6. Determine the points of intersection of r = cos(2 θ ) and r = 1 + cos θ. Remark 1.1. In the polar coordinate system, the same point can be represented by very many diﬀerent pairs of polar co-ordinates. Some care is therefore required in ﬁnding intersections. Graphs can be very helpful for checking results. 2...
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30 - x 2 y 2-4 x = 0 1.1 Curve Sketching Example 1.4 Trace...

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