Polar Coordinates Honors PreCalculus 9 1 The Polar

Polar Coordinates Honors Pre-Calculus 9. 1

The Polar Coordinate System • Origin: a fixed point O • Polar Axis: initial ray from the pole • Usually horizontal and directed toward the right • Polar Coordinates: (r, ϴ) • r is the directed distance from the pole • ϴ is the directed angle from the polar axis to OP

When Graphing the Polar Coordinates • Positive theta values rotate counter-clockwise; Negative theta clockwise • If r value is positive, then P lies on the terminal side of theta. If r is negative, P lies on the ray opposite the terminal side of theta.

Example 1: Graph the Polar Coordinates a) A(2, 45 o) b) B(-1. 5, ) c) C (3, -30 o) ***If Graphing on the Polar Grid, you are essentially graphing on the Unit Circle (radius doesn’t technically have to be one) ***

When Graphing the Polar Coordinates • Since angles have infinitely many co-terminal angles. As a result if a point has polar coordinates at (r, ϴ) then it also has polar coordinates at (r, ϴ +/- 360 o) or (r, ϴ +/- 2π). • Additionally, because r is a directed distance (r, ϴ) and (-r, ϴ +/- 180 o) or (-r, ϴ +/- π).

Example 2: • Point T is (4, 135 o) is a point on the Polar Grid. Find the other three representations of point T. (Assume -360 o ≤ ϴ ≤ 360 o)

Graphs of Polar Equations • Polar Equations: an equation expressed in terms of polar coordinates • Polar Graph: the set of all points with coordinates (r, ϴ) that satisfies a given polar equation. Example 3: a) r = 2 b) ϴ = 30 o

Polar Distance Formula • If P 1(r 1, ϴ 1) and P 2(r 2, ϴ 2) are two points, on the polar plane, The distance of P 1 P 2 is