10.1. Polar and Rectangular Coordinates http://www.ck12.org
This is the equation of a hyperbola.
Concept Problem Revisited
The general way to express a liney=mx+bin polar form isr=sinθ−bm·cosθ.
Vocabulary
Thepolar coordinate systemdefines each point by its angle on the unit circle(θ)and its distance from the origin
(r). Points in the polar coordinate system are written as(r,θ).
Therectangular coordinate systemdefines each point by its distance from thexandyaxes. Points in the rectangular
coordinate system are written as(x,y).
Guided Practice
- Sketch the following polar equation:r=3.
- Sketch the following polar equation:r=θwithθ: 0≤θ≤ 2 π.
- Translate the following polar expression into rectangular coordinates and then graph.
r= 2 ·sec(θ−π 2 )
Answers: - Since theta is not in the equation, it can vary freely. This simple equation produces a perfect circle of radius 3
centered at the origin.
You can show this equation is equivalent tox^2 +y^2 = 9 - The equationr=θis an example of a polar equation that cannot be easily expressed in rectangular form. In
order to sketch the graph, identify a few key points:( 0 , 0 ),(π 2 ,π 2 ),(π,π),(^32 π,^32 π),( 2 π, 2 π). You should see that the
shape is very recognizable as a spiral.