6.3 Polar Coordinates Day One Objectives Plot points in the polar coordinate system Find multiple sets of polar coordinate for a given point Convert a point from polar to rectangular coordinates Convert a point from rectangular to polar coordinates Pg. 655 #2-48 (even)

Defining points in the polar system Location of a point is based on radius (distance from the origin) and theta (the angle the radius moves from standard position (positive x-axis in a cartesian system) (r, Θ) is the point in polar coordinates. r and Θ can be positive, negative, or zero.

The point described in polar coordinates by (2, 3π/4) would look like this:                                                           1. Plot the point (3, 315o) on the polar graph provided.

2. Plot the point (2, 60°) 3. Plot the point (4, 165°)

4. Plot the point (-2, π) 5. Plot the point (-1, )

Multiple Representations of Points If n is an integer, then the point (r, Θ) can be represented as (r, Θ + 2nπ) or (-r, Θ + π + 2nπ) Find another representation of (5, ) in which r is positive and 2π < Θ < 4π r is negative and 0 < Θ < 2π r is positive and - 2π < Θ < 0

Converting from Polar Coordinates to Rectangular Coordinates If cos θ = x/r then x = r cos θ If sin θ = y/r then y = r sin θ Find the rectangular coordinates for the points given in polar coordinates. (3, π) 10. (-10, )

Converting from Rectangular Coordinates to Polar Coordinates From Pythagoras, we have: r2 = x2 + y2 and, thus, and basic trigonometry gives us:               Step 1: Determine the quadrant in which the point resides. Step 2: Find r by computing the distance from the origin to (x,y) using Step 3: Find Θ using . Make sure the angles fits the quadrant determined in step 1. Note: We are assuming that r > 0 and 0 < Θ < 2π

11. Find the polar coordinates of the point whose rectangular coordinates are (1, - )