polar coordinates

polar coordinates - CDC Jaurigue POLAR COORDINATES 1 THE...

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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 ) , ( r where r is the distance from O to P is the angle made from OA to OP . h If is positive, the rotation is counterclockwise h If 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 . Remark: A point on the polar plane can be represented by more than one ordered pair. That is, there is no one-to-one 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 ( A 5. ) 6 / , 5 ( - E 2. ) 4 / 5 , 3 ( B 6. ) 3 / , 6 ( -- F 3. ) 3 / 4 , 4 ( - C 7. ) , 7 ( G 4. ) 9 / , ( D 8. ) 2 / 3 , 8 ( H P 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 cos cos r x r x = = sin sin r y r y = = Example 2. Convert the following polar coordinates to Cartesian coordinates. 1. ) , 3 ( 2. - 3 2 , 4 Solution: 1. 3 ) 1 ( 3 cos 3 cos- =- = = = r x ) ( 3 sin 3 sin = = = = r y Therefore, the point ) , 3 ( in polar coordinates is equivalent to the point ) , 3 (- in Cartesian coordinates. 2. 2 ) 2 / 1 ( 4 ) 3 / 2 cos( 4 cos =-- =- = = r x 3 2 ) 2 / 3 ( 4 ) 3 / 2 sin( 4 sin- =- =- = = r y Therefore, the point - 3 2 , 4 in polar coordinates is the same as the point ) 3 2 , 2 (- in Cartesian coordinates....
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polar coordinates - CDC Jaurigue POLAR COORDINATES 1 THE...

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