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Polar Coordinates

# Polar Coordinates - Polar Coordinates Advanced Level Pure...

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Unformatted text preview: Polar Coordinates Advanced Level Pure Mathematics Advanced Level Pure Mathematics Calculus II Introduction 2 Relations between Cartesian and Polar Coordinates 3 Sketch of Graphs in Polar Coordinates 5 Intersection of Two Curves in Polar Coordinates 10 Slope of a Tangent 11 Areas 13 Arc Lengths 15 Surfaces of Revolution 16 Introduction Cartesian Coordinate plane Polar Coordinate plane Prepared by K. F. Ngai Page 1 Polar Coordinates Advanced Level Pure Mathematics y p (r, θ ) p (x, y) y vectorial angle θ x x 0 origin polor axis (pole) (initial axis) Example y P( 2 , 45 ° ) p (1, 1) 2 1 x 0 45 ° 1 N.B. (1) θ is positive if it is measured anti-clockwise . (2) θ is negative if it is measured clockwise . P (4, 6 π ) - 6 π 4 6 π Q (4, - 6 π ) (3) r can be negative. A ( 2 , 3 π ) A ( 2 , 3 π ) 2 2 3 π 3 π 2 0 B (- 2 , 3 π ) (4) If r > 0 , P ( r, θ ) = P ( r, 2n π + θ ) , 2200 n ∈ Z . e.g. P ( r, π ) = P ( r, 3 π ) = P ( r, - π ) p (r, 3 π ) p (r, π ) π r p (r, - π ) (5) If r = 0 , O ( 0, θ ) is called the pole or origin , 2200 θ ∈ R . (6) If r > 0 , Q ( -r, θ ) = Q ( r, π + θ ) . 3 π Prepared by K. F. Ngai Page 2 Polar Coordinates Advanced Level Pure Mathematics 4 Q (4, π + 3 π ) or Q (-4, 3 π ) (7) Q is an imaginary point (which r<0) and P is a real point (which r>0) . Relations between Cartesian and Polar Coordinates x r y r r x y y x = = ⇔ = + = - cos sin tan θ θ θ 2 2 1 y P (x, y) P (r, θ ) r y θ x (polar axis) 0 x Example Express the Cartesian coordinates in polar coordinates (a) ( , ) 2 3 2 , (b) ( , ) 2 2- (a) 4 4 12 2 2 = + = + = y x r 6 3 1 tan 3 2 2 tan 1 1 π θ = = =-- so the polar coordinates= ) 6 , 4 ( π (b) Example Express the polar coordinates in Cartesian coordinates (a) (-2 , π ) , (b) ( , ) 2 3 6- π Prepared by K. F. Ngai Page 3 Polar Coordinates Advanced Level Pure Mathematics Definition Polar Equations : r = f( θ ) Example r=2cos θ , r=2(1-sin θ ) , r=3 , etc. Example Investigate the loci represented by the following polar equations....
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Polar Coordinates - Polar Coordinates Advanced Level Pure...

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