Appendix H Part 1
Polar Coordinates
(1) Polar Coordinates and Cartesian Coordinates,
(2) Graphing Polar Curves,
(3) Derivatives of Polar Functions.
MATH 122 (Appendix H Part 1)
Polar Coordinates
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We usually use Cartesian coordinates (x, y ) to represent a point in a
plane. Newton introduced the polar coordinate system which is more
convenient for many purposes.
Fix a point O in a plane (called the
pole or origin) and a ray starting at
O, the polar axis.
If P is any point in the plane, let r be the distance from O to P and
let θ be the angle between OP and the polar axis.
Then the point P is represented by the ordered pair (r,θ), the
polar coordinates of P.
MATH 122 (Appendix H Part 1)
Polar Coordinates
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Convention:
θ is positive if it is measured counterclockwise from the polar axis to
OP
It is possible that r is negative. In this case, (−r , θ) = (r , θ + π).
Example:
Plot the points
π
P1 = 1,
4
π
P2 = −1,
4
MATH 122 (Appendix H Part 1)
P3 =
Polar Coordinates
7π
1,
4
π
P4 = 1, −
4
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Polar and Cartesian Coordinates
To convert from Polar to Cartesian coordinates, let
x = r cos(θ)
y = r sin(θ)
To convert from Cartesian to Polar coordinates, let
y r2 = x2 + y2
θ = arctan
x
Example
√
(I) The Polar point 4, π6 is equivalent to the Cartesian point (2 3, 2).
(II) The
√ Cartesian point
(3, −6) is equivalent to the Polar point
3 5, − arctan(2) .
MATH 122 (Appendix H Part 1)
Polar Coordinates
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Polar Curves, r = f (θ)
Sketch r = 3 and θ = − π3 .
Sketch r = 4 sin(θ).
(I) Use a calculator.
(II) Represent the function in Cartesian coordinates.
r = 4 sin(θ)
⇒
x 2 + (y − 2)2 = 4
(III) Graph r = 4 sin(θ) on a Cartesian plane with r and θ axes, then
convert to the Polar plane.
www.math.ku.edu/u/brennanj/Geogebra/CHS1
MATH 122 (Appendix H Part 1)
Polar Coordinates
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Tangents to Polar Curves
Polar curves that appear smooth do have tangent lines to their points. Let
r = f (θ).
The tangent line will pass through the Polar point (r0 , θ0 ) where
r0 = f (θ0 ), which is equivalent to (r0 cos(θ0 ), r0 sin(θ0 )).
The slope of the tangent line can be calculated using the Chain Rule
dy dy f 0 (θ) sin(θ) + f (θ) cos(θ) dθ
=
= 0
m=
dx dx (r0 ,θ0 )
f (θ) cos(θ) − f (θ) sin(θ) θ=θ0
dθ θ=θ0
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Polar Coordinates
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Example: Find the slope of the tangent line to the curve at any point and
then find the points on the curve where the tangent is horizontal or
vertical. r = 1 + cos(θ)
x = r cos(θ) = cos(θ) + cos2 (θ)
y = r sin(θ) = sin(θ) + cos(θ) sin(θ)
m=
−(2 cos(θ) − 1)(cos(θ) + 1)
sin(θ)(1 + 2 cos(θ))
Vertical
Tangents
4π
θ ∈ 0, 2π
3 , 3
Horizontal
Tangents
θ ∈ π3 , π, 5π
3
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