Warm-Up 12/05
Identify all angles that are coterminal with the given angle. Then give
one positive and one negative angle coterminal with the given angle.
1. 165°
165° + 360k°
525°; – 195°
2.
4𝜋
3
4𝜋
+ 2k°
3
10𝜋
2𝜋
;−
3
3
Rigor:
You will learn how graph points and simple
graphs with polar coordinates.
Relevance:
You will be able to use Polar Coordinates to solve
real world problems.
9-1 Polar Coordinates
Polar Coordinate System or polar plane
Pole is the origin
Polar axis is an initial ray from the pole.
Polar Coordinates (r,  )
r is directed distance from the pole
 is the directed angle from the polar axis.
Example 1: Graph each point.
a. A(2, 45°)
b. B(– 1.5,
2𝜋
)
3
c. C(3, – 30°)
Example 2: Graph points on a Polar Grid.
a. P(3,
4𝜋
)
3
b. Q(– 3.5, 150°)
In a rectangular coordinate system each point has a unique
set of coordinate. This is not true in a polar coordinate
system.
Example 3: Find four different pairs of polar coordinates that
name point T if – 360°≤  ≤ 360°.
(4, 135°)
(4, 135°) = (4, 135° – 360°) = (4, – 225°)
(4, 135°) = (– 4, 135° + 180°) = (–4, 315°)
(4, 135°) = (– 4, 135° – 180°) = (–4, – 45°)
Polar equation is an equation expressed in terms of
polar coordinates. For example, r = 2 sin.
Polar graph is the set of all points with coordinates
(r,  ) that satisfy a given polar equation.
Example 4: Graph each polar equation.
a. r = 2 (2, )

𝜋
4

4𝜋
3
r
2
2
2
b. 𝜃 =
𝜋
6
(r,
𝜋
)
6
r
– 3.5
1
4

𝜋
6
𝜋
6
𝜋
6
Example 5: Find the distance between the pair of points.
A(5, 310°), B(6, 345°)
𝐴𝐵 =
𝑟1 2 + 𝑟2 2 − 2𝑟1 𝑟2 cos 𝜃2 − 𝜃1
𝐴𝐵 =
52 + 62 − 2 5 6 cos 345° − 310°
𝐴𝐵 ≈ 3.4425
−1
math!
9-1 Assignment: TX p538, 2-42 even