1. Graphing Polar Coordinates
5015

2. Angle pos counter clockwise neg clockwise
Preliminaries:
Pole:
Polar Axis:
Coordinates:
r =
θ =
origin
Positive x-axis
Radius (r, θ)
Angle pos counter clockwise
neg clockwise

3. Plot the following points:
Degree Measure:
A(4, 330⁰) B)(3, -90⁰)
C(-2, 30⁰) D(-4, -45⁰) D C A B

4. Radian Measure:
A 3, 7𝜋 6 B 2, − 3𝜋 4
C −4, 𝜋 D −5, − 𝜋 3

5. Naming Polar Coordinates – all four ways
(4, 225)
4, −135
−4, −315
−4, 45

6. Examples: A(5, 150⁰) B(3, -240⁰)

7. Converting Polar Rectangular

8. Converting 𝑃(𝑟, 𝜃)→𝑅(𝑥, 𝑦)
Remember the unit circle:
𝑐𝑜𝑠𝜃, 𝑠𝑖𝑛𝜃
Now radius can be different values
(𝑡, 𝜃)
𝑠𝑖𝑛𝜃= 𝑦 𝑟
cos 𝜃= 𝑥 𝑟 y r 𝜃
𝑦=𝑟𝑠𝑖𝑛 𝜃
𝑟𝑐𝑜𝑠 𝜃=𝑥 x
Polar, 𝑃(𝑟,𝜃) rectangular, R(x,y)

9. TO CONVERT
Polar to Rectangular 𝑃(𝑟, 𝜃)→𝑅(𝑥, 𝑦)
𝑃(5, 60°)
𝑦=𝑟𝑠𝑖𝑛 𝜃
𝑥=𝑟𝑐𝑜𝑠 𝜃
𝑦=5 sin 60°
𝑥=5 cos 60° 𝑦=
𝑥= 5 2
𝑅 5 2 ,

10. 𝑃 4, 11𝜋 6
𝑥=𝑟𝑐𝑜𝑠 𝜃
𝑦=𝑟𝑠𝑖𝑛 𝜃
𝑥=4 cos 11𝜋 6
𝑦=4 sin 11𝜋 6
𝑥=4∗ =2 3
𝑦=4 − 1 2 =−2
𝑅 2 3 ,−2

11. Convert Rectangular to Polar 𝑅 𝑥, 𝑦 →𝑃(𝑟, 𝜃)+
𝑟= 𝑥 2 + 𝑦 2
𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥 -
If x is negative add 180 or 𝜋
If y only is negative add 2𝜋
𝑅(5, 9)
𝜃= 𝑡𝑎𝑛 −1 9 5 𝑟=
𝑟= 106
𝜃=61°
𝑃( 106 , 61°)

12. 𝑅(−6, −3)
𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥
𝑟= 𝑥 2 + 𝑦 2
𝜃= 𝑡𝑎𝑛 −1 −3 −6
𝑟= − −3 2
𝑟= 45
𝜃≈26.6
But we know the angle is in the III quadrant we will add 180°
𝑟=3 5
So 𝑃 3 5 ,206.6°

13. 𝑅(−5, 12)
𝑅(3, −4)
𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥
𝑟= 𝑥 2 + 𝑦 2
𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥
𝑟= 𝑥 2 + 𝑦 2
𝜃= 𝑡𝑎𝑛 −1 −4 3
𝑟= −4 2
𝜃= 𝑡𝑎𝑛 −1 12 −5
𝑟= −
𝑟=5
𝜃≈−53.1
𝑟=13
𝜃≈−67.4°
But we know the angle is in the IV quadrant so we will add 360°
We know the angle is in the II quadrant so we add 180°
𝑃 5, 306.9°
𝑃 13, 112.6°