10.6 Polar Coordinates

10.6 Polar Coordinates

The polar coordinate system is different from the rectangular system, which is what our class is used to thus far. The system starts at point "O", which is the origin, but in the polar system it is called the "pole". Points in the polar coordinate system are represented in the form of (r,θ), where r (radius)= distance from the pole to the point (P) and θ= the angle deviated from the polar axis.

 A polar coordinate system graph looks like this.

To plot a given point on the polar coordinate system:

1. Start at "O" or the pole

2. Jump r number of rings along polar axis

3. Move θ around the circle

The same coordinate can be represented in an infinite amount of ways. The radius can be positive or negative. 

 or

Converting coordinates from rectangular to polar is simple. To visualize, the polar axis is relatable to the x axis and the pole is relatable to the origin. 

           

   and

                    

When converting entire equations, replace the x, y, r, or θ with is corresponding conversion.

Solve for θ:

Take the inverse tangent of -1 to get:

Solve for r:

Final polar coordinate:

___________________________

Solve for x:

Solve for y:

Final rectangular coordinate:

The polar coordinate system is very relatable to the Unit Circle. When the radius is 1, it is the unit circle. Positive angles for θ always  go counterclockwise. Negative angles for θ always go clockwise.