9.6 – POLAR COORDINATES

I N THIS SECTION, YOU WILL LEARN TO  plot points in the polar coordinate system  convert points from rectangular to polar form and vice versa  convert equations from rectangular to polar form and vice versa

POLAR COORDINATE SYSTEM:  So far, you have been working in the rectangular coordinate system, where ( x, y ) represented the directed distances from the coordinate axes.  You will now be working in the polar coordinate system.

POLAR COORDINATE SYSTEM: a) Definition: A point P in the plane has polar coordinates if the line segment OP has length and the angle that OP makes with the positive axis is (measured in a counter clockwise direction). The fixed point O is called a pole and initial ray from O is called the polar axis.

POLAR COORDINATE SYSTEM: Polar Coordinate Directed angle Pole Polar Axis

POLAR COORDINATE SYSTEM:

   

C OORDINATE C ONVERSION :  Rectangular coordinates can be converted to polar coordinates and vice versa.  Then the polar coordinates and the cartesian coordinates ( x,y ) of the same point are related as follows:

P OLAR TO R ECTANGULAR C OORDINATES :  To convert between polar and rectangular coordinates, we make a right triangle to the point ( x,y ) like this:

P OLAR TO R ECTANGULAR C OORDINATES :

R ECTANGULAR TO P OLAR C OORDINATES : 

C OORDINATE C ONVERSION :

C ONVERTING P OLAR E QUATIONS TO R ECTANGULAR E QUATIONS : 

C ONVERTING R ECTANGULAR E QUATIONS TO P OLAR E QUATIONS : When you graph this on the polar system, it is a circle with radius 3. Therefore, the rectangular equation should also reflect a circle with radius 3.

C ONVERTING R ECTANGULAR E QUATIONS TO P OLAR E QUATIONS : When you graph this on the polar system, it is line at this angle.

C ONVERTING R ECTANGULAR E QUATIONS TO P OLAR E QUATIONS :