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, ) p (x, y) y vectorial angle 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) is positive if it is measured anti-clockwise . (2) is negative if it is measured clockwise . P (4, 6 ) - 6 4 6 Q (4, - 6 ) (3) r can be negative. A ( 2 , 3 ) A ( 2 , 3 ) 2 2 3 3 2 0 B (- 2 , 3 ) (4) If r > 0 , P ( r, ) = P ( r, 2n + ) , 2200 n Z . e.g. P ( r, ) = P ( r, 3 ) = P ( r, - ) p (r, 3 ) p (r, ) r p (r, - ) (5) If r = 0 , O ( 0, ) is called the pole or origin , 2200 R . (6) If r > 0 , Q ( -r, ) = Q ( r, + ) . 3 Prepared by K. F. Ngai Page 2 Polar Coordinates Advanced Level Pure Mathematics 4 Q (4, + 3 ) or Q (-4, 3 ) (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 2 2 1 y P (x, y) P (r, ) r y 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 = = =-- so the polar coordinates= ) 6 , 4 ( (b) Example Express the polar coordinates in Cartesian coordinates (a) (-2 , ) , (b) ( , ) 2 3 6- Prepared by K. F. Ngai Page 3 Polar Coordinates Advanced Level Pure Mathematics Definition Polar Equations : r = f( ) Example r=2cos , r=2(1-sin ) , 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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