Rectangular (Cartesian) coordinates plot a point by moving left/right and up/down (making a rectangle)

Polar coordinates find the same point in a different way

r = distance from the origin (radius)θ = angle with positive x-axis

Polar Coordinates

Ex. The Cartesian coordinates of a point are given, find the polar coordinates.

a) (0,1)

b)

c)

2 22 2,

2 3,2

cos

sin

x r

y r

2 2 2

tan yx

x y r

Polar Rect Rect Polar

Ex. The polar coordinates of a point are given, find the Cartesian coordinates.

a) (2,π)

b) 63,

Ex. Find the polar equation for the curve.a) x2 + y2 = 3y

b) x3y2 + ln y = 3

Ex. Find the Cartesian equation for the curve.a) r = cos θ

b) sin θ = r2 cos θ

Ex. Sketch the polar equation r = 3

6

4

2

- 2

- 4

- 6

- 5 5

Ex. Sketch the polar equation θ =

6

4

2

- 2

- 4

- 6

- 5 5

3

Ex. Sketch the polar equation r = sec θ.

6

4

2

- 2

- 4

- 6

- 5 5

Ex. Sketch the polar equation r = 2cos5θ

6

4

2

- 2

- 4

- 6

- 5 5

For the function r = f (θ),

cos sin

cos sin

dyd

dxd

f fdy

dx f f

Ex. Find the points of horizontal and vertical tangency on the graph r = sin θ, 0 ≤ θ ≤ 2.

Pract.

1. Convert the polar point to Cartesian.

2. Sketch the curve r = 2cos θ.

3. Find the slope of the tangent line to r = 1 + sin θ when θ =

1, 3

32,

13