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Parametric Equations and Polar Coordinates

Problems

Graphing in Polar Coordinates

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Problem : Do the rules of symmetry show that the following polar graph is symmeric with respect to either the pole, the polar axis, or the line θ = ?

a) r = cos(θ) + 2 .
b) r = 2 sin(θ) .
c) r = 7 .
d) r = 2 cos(3θ) ; e) r = .

a) r = cos(θ) + 2 is symmetric with respect to the polar axis because cos(θ) + 2 = cos(- θ) + 2 .
b) r = 2 sin(θ) is symmetric with respect to the line θ = because 2 sin(θ) = 2 sin(Π - θ) .
c) r = 7 is a circle -- it is symmetric with respect to the pole, the polar axis, and the line θ = .
d) r = 2 cos(3θ) is symmetric with respect to the polar axis because 2 cos(3θ) = 2 cos(- 3θ) .
e) r = is symmetric with respect to the pole, because 2 sin(2θ) = 2 sin(2(θ + Π)) .

Problem : What is the maximum value of | r| for the following polar equations: a) r = cos(2θ) ; b) r = 3 + sin(θ) ; c) r = 2 cos(θ) - 1 .

a) The maximum value of | r| in r = cos(2θ) occurs when θ = where n is an integer and | r| = 1 .
b) The maximum value of | r| in r = 3 + sin(θ) occurs when θ = +2 where n is an integer and | r| = 4 .
c) The maximum value of | r| in r = 2 cos(θ) - 1 occurs when θ = (2n + 1)Π where n is an integer and | r| = 3 .

Problem : Find the intercepts and zeros of the following polar equations: a) r = cos(θ) + 1 ; b) r = 4 sin(θ) .

a) Polar axis intercepts: (r, θ) = (2, 2),(0,(2n + 1)Π) , where n is an integer. Line θ = intercepts: (r, θ) = (1, + ) , where n is an integer. r = cos(θ) + 1 = 0 for θ = (2n + 1)Π , where n is an integer.
b) Polar axis intercepts: (r, θ) = (0, ) where n is an integer. Line θ = intercepts: (r, θ) = (4, +2) where n is an integer. r = 4 sin(θ) = 0 for θ = , where n is an integer.

Problem : Decide whether each of the following polar graphs is a limacon, a rose curve, a spiral, a circle, or none of these: a) r = 2 + cos(θ) ; b) r = 2 ; c) r = sin(3θ) ; d) r = 1 - cos(θ) ; e) r = 2θ .

a) r = 2 + cos(θ) is a limacon.
b) r = 2 is a circle.
c) r = sin(3θ) is a rose curve.
d) r = 1 - cos(θ) is a limacon.
e) r = 2θ is a spiral.

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