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polar curves

# polar curves - Curves in Polar Coordinates Math 213 Fall...

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Curves in Polar Coordinates Math 213 — Fall 2007 You should immediately memorize the following formulas: x = r cos θ r 2 = x 2 + y 2 y = r sin θ tan θ = y/x In the following, we allow r and θ to take arbitrary values. Your textbook and instructor may use diﬀerent conventions. For ex- ample, it is common to assume that r 0 and 0 θ < 2 π . A polar curve can usually be described by an implicit equation F ( r,θ ) = 0 for some regular function F . 1 It is often the case that a curve can be described in the form r = f ( θ ) , i.e. F ( r,θ ) = r - f ( θ ). More generally, many polar curves can be written in the form g ( r ) = f ( θ ) , i.e. F ( r,θ ) = g ( r ) - f ( θ ). In the sequel, we will mostly focus on curves of the form r = f ( θ ). Tests for Symmetry You can use the following algebraic tests to check for symmetries. The graph of the polar equation r = f ( θ ) has: 1 In this context, “regular” means that the partial derivatives ∂F/∂r and ∂F/∂θ are never both 0 when F ( r,θ ) = 0. 1

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Symmetry about the x -axis when θ can be replaced by - θ everywhere without changing the equation, i.e. f ( θ ) = f ( - θ ). Example r = 2 cos θ . Symmetry about the
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polar curves - Curves in Polar Coordinates Math 213 Fall...

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