Polar Coordinates - Anderson School District Five...The point P in the polar coordinate system is...
Transcript of Polar Coordinates - Anderson School District Five...The point P in the polar coordinate system is...
Polar Coordinates
The foundation of the polar coordinate system is a horizontal ray that extends to the right.
The ray is called the polar axis.
The endpoint of the ray is called the pole.
The point P in the polar coordinate system is represented by an ordered pair of numbers π, π .
r is a directed distance from the pole to P. (it can be positive, negative, or zero.)
π is an angle from the polar axis to the line segment from the pole to P.
The Sign of r and a Pointβs Location in Polar Coordinates
The point π = π, π is located π units from the pole.
-- π > 0, the point lies along the terminal n
side of π.
-- π < 0, the point lies along the ray opposite
the terminal side of π.
-- π = 0, the point lies at the pole, regardless
of π.
Plot the point.
(2, 135Β°)
Plot the point.
(β3,3π
2)
Plot the point.
β1,βπ
4
Multiple Representations of Points.
If n is any integer, the point (π, π) can be represented as
π, π = (π, π + 2ππ) or
π, π = (βπ, π + π + 2ππ)
Find 3 representations
(2,π
3)
a. r is positive and 2π < π < 4π
b. r is negative and 0 < π < 2π
c. r is positive and β2π < π < 0
Find 3 representations
(5,π
4)
a. r is positive and 2π < π < 4π
b. r is negative and 0 < π < 2π
c. r is positive and β2π < π < 0
Graph of a Circle
π = 2
Graph of a Line
π =π
6
Relations between Polar and Rectangular Coordinates
π₯2 + π¦2 = π2
r π πππ =
y πππ π =
ΞΈ π‘πππ =
x
π₯ =
π¦ =
π‘πππ=
Find the rectangular coordinates
(2,3π
2)
Find the rectangular coordinates:
β8,π
3
What if itβs not on the Unit Circle?
(3, 52Β°)
Not on Unit Circle
(4, β168Β°)
Converting a Point from Rectangular to Polar Coordinates
1. Plot the point (π₯, π¦).
2. Find r by computing the distance from
the origin to π₯, π¦ : π = (π₯2 + π¦2).
3. Find π using π‘πππ =π¦
π₯ with the terminal
side of π passing through (π₯, π¦).
Find the Polar Coordinates
β1, 3
Find the Polar Coordinates
(1, β 3)
Find the polar coordinates
(0, -5)
Not on Unit Circle
(β3, 2)
Not on Unit Circle
(β4, 7)
One More
(4, β6.2)
Equation Conversion from Polar to Rectangular
Use one or more of these equations:
π2 = π₯2 + π¦2 ππππ π = π₯ ππ πππ = π₯ π‘πππ =π₯
π¦
Convert to a rectangular equation: π = 10
Convert to a rectangular equation
π =π
3
Convert to a rectangular equation
ππππ π = 7
Convert to a rectangular equation
π = 6π πππ
Convert to a rectangular equation
π = 8πππ ππ + 2π πππ
Convert to a rectangular equation
π2π ππ2π = 4
Equation Conversion from Rectangular to Polar
To convert a rectangular equation in x and y to a polar equation that expresses r in terms of π:
--replace x with ππππ π
--replace y with ππ πππ
Convert to a polar equation
π₯ + 5π¦ = 8
Convert to a polar equation
π¦ = 3
Convert to a polar equation
π₯2 + π¦2 = 16
Convert to a polar equation
π₯2 + π¦ + 3 2 = 9
Convert to a polar equation
π₯2 = 6π¦