6.3 Polar CoordinatesDay 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.

P = (r, )

r

polar axisoriginpoint

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, ) 2

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

6. r is positive and 2π < Θ < 4π

7. r is negative and 0 < Θ < 2π

8. r is positive and - 2π < Θ < 0

4

4

Converting from Polar Coordinates to Rectangular Coordinates

If cos θ = x/r then x = r cos θ If sin θ = y/r theny = r sin θ

Find the rectangular coordinates for the points given in polar coordinates.

9. (3, π) 10. (-10, ) 6

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

2 2r x y

2 2r x y

11. Find the polar coordinates of the point whose rectangular coordinates are (1, - )3

12. Find the polar coordinates of the point whose rectangular coordinates are (0, -4)