Polar Coordinates KUS objectives BAT relate Cartesian and

Polar Coordinates • 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

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 Polar coordinates describe equivalent points, but in a different way Polar coordinates use the distance from the origin, and the angle from the positive x-axis 1 (-4, -1) Polar coordinates – use the distance and the angle (5, 0. 93) 5 3. 39 c 0. 93 c RADIANS will be most commonly used in this chapter Anglesd can be positive (CCW) or negative (CW) GEOGEBRA 4. 2 (4. 2, 3. 39)

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

WB 1 Practice some points Try these:

WB 2 a) Find the Polar coordinates of the following point: (5, 9) (x, y) Hyp We can use Trigonometry and Pythagoras to link Cartesian and Polar coordinates r y Opp θ x Sub in Adj and Hyp Sub in Opp and Hyp Cartesian Polar Sub in Opp and Adj Tan-1 (also known as arctan) Adj Sub in r, x and y

WB 2 b) Find the Polar coordinates of the following points: (5, -12) a) Draw a diagram 5 θ r 12 (5, -12) Cartesian Polar Notice the angle is negative, as we have measured it the opposite way (clockwise)

Draw a diagram 1 √ 3 θ r (√ 3, -1) Cartesian Polar 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!)

Use Radians WB 2 Cartesian to Polar coordinates II (12, 5) R and θ? (-3, 0) (2, -2)

π/ 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 2 5 π 5√ 3 π/ 0 3 10 (10, 4π/3) So the Cartesian coordinate is (-5, -5√ 3) (remember to interpret whether values should be negative or positive from the diagram!) 3π/ 2

π/ (8, 2π/3) 4√ 3 π So the Cartesian coordinate is (-4, 4√ 3) (remember to interpret whether values should be negative or positive from the diagram!) 2 8 π/ 3 0 4 3π/ 2

WB 3 Polar to Cartesian coordinates. I x and y? Use Radians

Practice Now do these questions SK 203_101

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

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