1. Section 6.3 Polar Coordinates

2. The foundation of the polar coordinate system is a horizontal ray that extends to the right. This ray is called the polar axis. The endpoint of the polar axis is called the pole. A point P in the polar coordinate system is designated by an ordered pair of numbers (r, θ). r pole polar axis θ P = (r, θ) r is the directed distance form the pole to point P ( positive, negative, or zero). θ is angle from the pole to P (in degrees or radians).

3. Plotting Points in Polar Coordinates.

4. Plot each point (r, θ) a) A(3, 45 0 ) A b) B(-5, 135 0 ) B c) C(-3, -π/6) C

5. CONVERTING BETWEEN POLAR AND RECTANGULAR FORMS CONVERTING FROM POLAR TO RECTANGULAR COORDINATES. To convert the polar coordinates (r, θ) of a point to rectangular coordinates (x, y), use the equations x = rcosθ and y = rsinθ

6. Convert the polar coordinates of each point to its rectangular coordinates. a) (2, -30 ⁰ ) b) (-4, π/3) a) x = rcos(-30 ⁰) b) x= -4cos(π/3) = -4(1/2) = -2 y= -4 sin(π/3) =

7. CONVERTING FROM RECTANGULAR TO POLAR COORDINATES: To convert the rectangular coordinates (x, y) of a point to polar coordinates: 1)Find the quadrant in which the given point (x, y) lies. 2) Use r =

8. Find the polar coordinates (r, θ) of the point P with r > 0 and 0 ≤ θ ≤ 2π, whose rectangular coordinates are (x, y) = The point is in quadrant 2. tanθ = The required polar coordinates are (2, 2π/3)

9. Give polar coordinates for the point shown.

10. EQUATION CONVERSION FROM RECTANGULAR TO POLAR COORDINATES. A polar equation is an equation whose variables are r and θ. Examples are To convert a rectangular coordinate equation in x and y to a polar equation in r and θ, replace x with rcosθ and y with rsinθ.

11. Example: Convert each rectangular equation to a polar equation that expresses r in terms of θ’ a)x + y = 5 b) ans. r= 2cosθ

12. EQUATION CONVERSION FROM POLAR TO RECTANGULAR COORDINATES.