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Unformatted text preview: CDC Jaurigue POLAR COORDINATES 1 THE POLAR COORDINATE SYSTEM Polar coordinates consist of a directed distance and the measure of an angle relative to a fixed point ( pole ) and a fixed ray ( polar axis ). • The point O is called the pole. • The ray OA is called the polar axis. • A point P has polar coordinates ) , ( θ r where r is the distance from O to P θ is the angle made from OA to OP . h If θ is positive, the rotation is counterclockwise h If θ is negative, rotation is clockwise. s If r is positive, count r units on the line representing the given angle. s If r is negative, count r units on the opposite side/direction of the given angle. s If = r , the point corresponds to the pole (or the origin) for whatever value of θ. Remark: A point on the polar plane can be represented by more than one ordered pair. That is, there is no onetoone correspondence between the set of points on the plane and the ordered pairs. Example 1. The following points are located on the polar plane below. 1. ) 6 / , 2 ( π A 5. ) 6 / , 5 ( π E 2. ) 4 / 5 , 3 ( π B 6. ) 3 / , 6 ( π F 3. ) 3 / 4 , 4 ( π C 7. ) , 7 ( G 4. ) 9 / , ( π D 8. ) 2 / 3 , 8 ( π H P Α Ο θ CDC Jaurigue 2 RELATIONSHIP BETWEEN CARTESIAN AND POLAR From the figure on the left side, we have the following relationships: 2 2 2 2 2 y x r r y x + = ⇒ = +  ⇒ = x y x y 1 tan tan θ θ θ θ cos cos r x r x = ⇒ = θ θ sin sin r y r y = ⇒ = Example 2. Convert the following polar coordinates to Cartesian coordinates. 1. ) , 3 ( π 2.  3 2 , 4 π Solution: 1. 3 ) 1 ( 3 cos 3 cos = = = = π θ r x ) ( 3 sin 3 sin = = = = π θ r y Therefore, the point ) , 3 ( π in polar coordinates is equivalent to the point ) , 3 ( in Cartesian coordinates. 2. 2 ) 2 / 1 ( 4 ) 3 / 2 cos( 4 cos = = = = π θ r x 3 2 ) 2 / 3 ( 4 ) 3 / 2 sin( 4 sin = = = = π θ r y Therefore, the point  3 2 , 4 π in polar coordinates is the same as the point ) 3 2 , 2 ( in Cartesian coordinates....
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This note was uploaded on 12/08/2011 for the course ECON 101 taught by Professor Smith during the Spring '11 term at Adrian College.
 Spring '11
 SMITH

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