1. Warm Up
Find magnitude and direction of each angle. Round your answers to the nearest tenth.
1. r = < 4, -5 >
2. v = < -3, -8 >
3. t = < -5, -9 >

2. Unit 6.1: Polar Coordinates

3. Polar Coordinates
The polar coordinate system records locations using distances and angles.
The origin is called the pole.
The polar axis is the initial ray from the pole directed horizontally toward the right
A point P in polar coordinates is of the form (𝑟,𝜃), where r is the directed distance from the pole to the point, and 𝜃 is the directed angle from the polar axis to 𝑂𝑃
r is the magnitude, and 𝜃 direction

4. Graph each point.
1. 𝐴( 2,45 𝑜 )
2. 𝐵(−1.5, 2𝜋 3 )
3. 𝐶(3,− 30 𝑜 )
4. 𝐷(−1, 𝜋 2 )
5. 𝐸( 2.5,240 𝑜 )
6. 𝐹(4,− 5𝜋 6 )

5. Can’t stay off the grid Graph each of point on a polar grid.
1. 𝑃(3, 4𝜋 3 )
2. 𝑄(− 3.5,150 𝑜 )
3. 𝑅(1.5,− 7𝜋 6 )
4. 𝑆(−2,− 135 𝑜 )

6. Coterminal Angles
Due to the existence of coterminal angles, the location of polar coordinates are not unique. As such, a point 𝑃(𝑟,𝜃) can have polar coordinates involving all coterminal angles of 𝜃.
As such, 𝑃(𝑟,𝜃) also has coordinates (𝑟,𝜃±360) or (𝑟,𝜃±2𝜋)

7. Other Directions Recall that r is a directed distance
-r would go in the opposite direction of r
As such, (𝑟,𝜃) can also be represented by 𝑟,𝜃±𝜋 or (𝑟,𝜃± 180 𝑜 )

8. Potato, potahto
Find four different pairs of polar coordinates that name point T if − 360 𝑜 ≤𝜃≤ 360 𝑜
Find three additional pairs of polar coordinates that name the given point below if − 360 𝑜 ≤𝜃≤ 360 𝑜 or −2𝜋≤𝜃≤2𝜋
1. ( 5,240 𝑜 ) 2. (2, 𝜋 6 )

9. Expressing Yourself
An equation expressed in terms of polar coordinates is called a polar equation
𝑟=2 sin 𝜃
A polar graph is the set of all points with coordinates 𝑟,𝜃 that satisfy a given polar equation
Just as graphs like x = 2 or y = -3 are constant on the Cartesian coordinate system, graphs of the polar equations r = k or 𝜃=𝑘 are also constant. As such, these graphs are considered basic in the polar coordinate system by mathematicians.

10. Graphs
Graph each polar equation.
1. 𝑟=2
2. 𝜃= 𝜋 6
3. 𝑟=3
4. 𝜃= 2𝜋 3

11. Polar Distance
If 𝑃 1 ( 𝑟 1 , 𝜃 1 ) and 𝑃 2 ( 𝑟 2 , 𝜃 2 ) are two points in the polar plane, then the distance 𝑃 1 𝑃 2 is given by 𝑟 𝑟 2 2 −2 𝑟 1 𝑟 2 cos⁡( 𝜃 2 − 𝜃 1 )

12. Don’t call me Shirley
An air traffic controller is tracking two airplanes that are flying at the same altitude. The coordinates of the planes are 𝐴( 5,310 𝑜 ) and 𝐵( 6,345 𝑜 ), where the directed distance is measured in miles.
Sketch the graph of this situation.
If regulations prohibit airplanes from passing within three miles of each other, are these airplanes in violation? Explain.

13. Bogey Inbound!
A naval radar is tracking two aircraft carriers. The coordinates of the two carriers are ( 8,150 𝑜 ) and ( 3,65 𝑜 ), with r measured in miles.
Sketch a graph of this situation.
What is the distance between the two aircraft carriers?