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10.3 Polar Coordinates

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One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. Initial ray A polar coordinate pair determines the location of a point. r – the directed distance from the origin to a point Ө – the directed angle from the initial ray (x-axis) to ray OP.

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(Circle centered at the origin) (Line through the origin) Some curves are easier to describe with polar coordinates: (Ex.: r = 2 is a circle of radius 2 centered around the origin) (Ex. Ө = π /3 is a line 60 degrees above the x-axis extending in both directions)

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More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are: Each point can be coordinatized by an infinite number of polar ordered pairs.

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Tests for Symmetry: x-axis: If (r, ) is on the graph,so is (r, - ).

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Tests for Symmetry: y-axis: If (r, ) is on the graph,so is (r, - )or (-r, - ).

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Tests for Symmetry: origin: If (r, ) is on the graph,so is (-r, )or (r, + ).

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Tests for Symmetry: If a graph has two symmetries, then it has all three:

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Try graphing this. (Pol mode)

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SPECIAL GRAPHS Circles: r = a cosθ r = a sinθ Lemniscates: r 2 = a 2 sin(2θ) r 2 = a 2 cos(2θ) Limaçons: r = a ± b(cosθ) r = a ± b(sinθ) a > 0, b > 0 Types of Limaçons: If, limaçon has an inner loop If, limaçon called a cardiod (heart shaped) If, limaçon with a dimple.

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SPECIAL GRAPHS Types of Limaçons: If, limaçon has an inner loop If, limaçon called a cardiod (heart shaped) If, limaçon with a dimple. If, convex limaçon.

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SPECIAL GRAPHS Rose curves: r = a cos(nθ) r = a sin(nθ) If n is odd, the rose will have n petals. If n is even, the rose will have 2n petals.

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CONVERTING TO RECTANGULAR COORDINATES: 1.) x = r cos Θy = r sinΘ 2.)

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Example: Convert the point represented by the polar coordinates (2, π) to rectangular coordinates. x = r cos(θ) x = 2cos(π) x = –2 y = r sin(θ) y = 2 sin(π) y = 0 So, (–2, 0)

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Example: Convert the point represented by the rectangular coordinates (–1, 1) to polar coordinates.

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Converting Polar Equations You can convert polar equations to parametric equations using the rectangular conversions. Example:

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Homework Section 10.4 –#1, 3, 11, 13, 23, 25, 27, 29, 31, 34, 35, 37, 41

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