Today in Precalculus

1. Today in Precalculus • Notes: Polar Coordinates (there is a handout) • Go over test • Homework

2. Polar Coordinate System P(r,θ) - a plane with a point O (the pole), and a ray from O (the polar axis). Each point P in the plane is assigned polar coordinates where r is the directed distance from O to P and θ is the directed angle whose initial side is on the polar axis and whose terminal side is on the line OP. θ O Polar axis

3. To graph the angle of a polar coordinate • If θ >0, it is measured counter-clockwise from the polar axis. • If θ <0, it is measured clockwise from the polar axis. • To graph the directed distance of a polar coordinate • If r >0, then P is on the terminal side of θ. • If r <0, then P is on the terminal side of θ+π

4. Example 1:

5. Example 2:

6. With coterminal angles, any point on the polar coordinate system can have infinitely many polar coordinates to describe this point. Example: Plot and rename the following with at least two different polar coordinates (2.5,30°) (2.5, 390°) (-2.5, 210°)

7. Example: Plot and rename the following with at least two different polar coordinates (-4, 100°) and (-4, 460°) (4, 280°)

8. In general, if (r, θ) is a point on the polar coordinate system so is (r, θ + 2 π) and (–r, θ + π). In fact, for any integer, n, (r, θ + 2πn) and (–r, θ + (2n+1)π) are all polar coordinates of the same point. • Examples: Find all of the polar coordinates of the given point, P. a) b) (-2, 75°) (-2, 75°+360°n) (2, 255°+360°n)