1. Polar
Coordinates

2. Polar Coordinates Starter: KUS objectives
BAT relate Cartesian and Polar forms of coordinates
Starter:
A plane is at a height of 200 m and a distance of 620 m East from a control tower.
Calculate the angle of elevation of the plane
Calculate the distance from the plane to the control tower
( Staying in the vertical plane between the plane and the tower )
The plane is flying at an angle of elevation of 160 to the horizontal at a speed of 30 𝑚𝑠 −1 . After 10 seconds
Calculate the new angle of elevation of the plane
Calculate the new distance from the plane to the control tower

3. Polar coordinates describe equivalent points, but in a different way
Notes 1
Cartesian coordinates – use horizontal and vertical position
(3,4)
The Cartesian way of describing coordinates uses x and y as the horizontal and vertical distances from the origin 4 4 3 1
(-4,-1)
Polar coordinates describe equivalent points, but in a different way
Polar coordinates – use the distance and the angle
Polar coordinates use the distance from the origin, and the angle from the positive x-axis
(5, 0.93) 5
3.39c
0.93c
RADIANS will be most commonly used in this chapter
Anglesd can be positive (CCW) or negative (CW)
4.2
(4.2, 3.39)
Negative values of r are shown by dotted lines but
For EDEXCEL 𝑟>0, so do not sketch parts with negative r
The range of 𝜃 is 0, 2𝜋
GEOGEBRA

4. We use radians as the units for direction
Notes 2
You can use either a set of Polar axes as shown or ordinary cartesian axes
We use radians as the units for direction

5. WB1 Practice some points
Try these:
2, 𝜋 2
−1, 𝜋 3
1, 𝜋
−2, 4𝜋 3
3, 3𝜋 4
1.5, − 𝜋 2
0.5, 7𝜋 6
0.5, − 5𝜋 6
2.5, 11𝜋 6
−1, − 𝜋 6

6. 𝑟 2 = 5 2 + 9 2 𝑟=10.3 𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 9 5 𝜃=60.9° (5,9) (10.3, 60.9°)
WB 2
a) Find the Polar coordinates of the following point: (5, 9)
(x,y)
Hyp
Opp r
We can use Trigonometry and Pythagoras to link Cartesian and Polar coordinates y θ x
𝑇𝑎𝑛𝜃= 𝑂𝑝𝑝 𝐴𝑑𝑗
𝐴𝑑𝑗=𝑐𝑜𝑠𝜃×𝐻𝑦𝑝
Adj
Sub in Adj and Hyp
Sub in Opp and Adj
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑇𝑎𝑛𝜃= 𝑦 𝑥
Tan-1 (also known as arctan)
𝑐 2 = 𝑎 2 + 𝑏 2
𝑂𝑝𝑝=𝑠𝑖𝑛𝜃×𝐻𝑦𝑝
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
Sub in r, x and y
Sub in Opp and Hyp
𝑟 2 = 𝑥 2 + 𝑦 2
𝑦=𝑟𝑠𝑖𝑛𝜃
𝑟 2 =
𝑟=10.3
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 9 5
𝜃=60.9°
(5,9)
(10.3, 60.9°)
Cartesian
Polar

7. b) (5, -12) (5,−12) (13, −67.4°) Cartesian Polar
WB 2 Find the Polar coordinates of the following points:
b) (5, -12)
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑦=𝑟𝑠𝑖𝑛𝜃
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
𝑟 2 = 𝑥 2 + 𝑦 2
a) Draw a diagram
𝑟 2 = 5
𝑟=13 θ 12 r
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 12 5
(5,-12)
𝜃=67.4°
(5,−12)
(13, −67.4°)
Cartesian
Polar
Notice the angle is negative, as we have measured it the opposite way (clockwise)

8. WB 2 Find the Polar coordinates of the following points:
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑦=𝑟𝑠𝑖𝑛𝜃
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
𝑟 2 = 𝑥 2 + 𝑦 2
Draw a diagram
𝑟 2 = ( 3 ) √3
𝑟=2 θ 1
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 r
(√3,-1)
𝜃= 𝜋 6
Notice we added π to the angle, so it would be in the correct quadrant (π/6 on its own when measured clockwise would not be in the right place!)
Cartesian
Polar
2, 7𝜋 6
(− 3 ,−1)

9. Use Radians
WB2 Cartesian to Polar coordinates II
(12, 5)
(-3, 0)
(2, -2)
R and θ?
2 2 , 𝜋
13, 0.394
3, 𝜋
2 2 , −1 4 𝜋
4, 3 2 𝜋
5, 4.07
4, −1 2 𝜋
2, 𝜋
5, −2.21

10. a) 10, 4𝜋 3 So the Cartesian coordinate is (-5,-5√3)
WB 3 Find the Cartesian coordinates of the following points:
a) 10, 4𝜋 3
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑦=𝑟𝑠𝑖𝑛𝜃
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
𝑟 2 = 𝑥 2 + 𝑦 2
c) As usual draw a diagram, and think carefully about which quadrant this point is in
A half turn would be π, and a 3/4 turn would be 3π/2, so this will be between those
π/2
𝑥=10𝑐𝑜𝑠 𝜋 3 5 π
𝑥=5
π/3
5√3
𝑦=10𝑠𝑖𝑛 𝜋 3 10
(10,4π/3)
𝑦=5 3
3π/2
So the Cartesian coordinate is
(-5,-5√3)
(remember to interpret whether values should be negative or positive from the diagram!)

11. b) 8, 2𝜋 3 So the Cartesian coordinate is (-4,4√3)
WB 3 Find the Cartesian coordinates of the following points:
b) 8, 2𝜋 3
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑦=𝑟𝑠𝑖𝑛𝜃
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
𝑟 2 = 𝑥 2 + 𝑦 2
π/2
(8,2π/3)
𝑥=8𝑐𝑜𝑠 𝜋 3 8
𝑥=4
4√3
π/3
𝑦=8𝑠𝑖𝑛 𝜋 3 π 4
𝑦=4 3
So the Cartesian coordinate is (-4,4√3)
3π/2
(remember to interpret whether values should be negative or positive from the diagram!)

12. Use Radians
WB3 Polar to Cartesian coordinatesI
x and y?
− 3 , 1
−2 2 , −2 2
0, 10
−3, −3 3
−5, 0
−1.25, 2.73

13. Practice
Now do these questions
SK 203_101

14. One thing to improve is –
KUS objectives BAT relate Cartesian and Polar forms of coordinates
self-assess
One thing learned is –
One thing to improve is –

15. END