1. 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)

2. 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.

3. 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.

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

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

6. 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

7. 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, )

8. 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

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