rectangular (cartesian) coordinates plot a point by moving left/right and up/down (making a...
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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