10.3 Polar Coordinates

A polar coordinate pair 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. A polar coordinate pair determines the location of a point. Initial ray r – the directed distance from the origin to a point Ө – the directed angle from the initial ray (x-axis) to ray OP.

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

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.

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

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

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

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

Try graphing this. (Pol mode)

SPECIAL GRAPHS Circles: Lemniscates: Limaçons: r = a cosθ r2 = a2sin(2θ) r = a ± b(cosθ) r = a sinθ r2 = a2cos(2θ) r = a ± b(sinθ) Types of Limaçons: a > 0, b > 0 If , limaçon has an inner loop If , limaçon called a cardiod (heart shaped) If , limaçon with a dimple.

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.

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.

CONVERTING TO RECTANGULAR COORDINATES: 1.) x = r cosΘ y = r sinΘ 2.)

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

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

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

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