Equivalent Polar Curves
The expression “same only different” comes into play in this section. We will graph two distinct polar equations that will produce two equivalent graphs. Use your graphing calculator and create these curves as the equations are presented.
Comparing Graphs of Polar Equations
Plot the following polar equations and compare the graphs.
Both equations, one in rectangular form and one in polar form, are circles with a radius of 4 and center at the origin.
Earlier, you were asked if there is a way you can determine if the two equations are equivalent.
As you learned in this section, we can compare graphs of equations to see if the equations are the same or not.
As you can see from the plots, your friend is correct. Your graph and his are the same, therefore the equations are equivalent.
For each equation in rectangular form given below, write the equivalent equation in polar form.
x2+y2=4 x2+y2=6y (x−1)2+y2=1 (x−4)2+(y−1)2=17 x2+y2=9
For each equation below in polar form, write another equation in polar form that will produce the same graph.
r=4+3sinθ r=2−sinθ r=2+2cosθ r=3−cosθ r=2+sinθ
Determine whether each of the following sets of equations produce equivalent graphs without graphing.
r=3−sinθand r=3+sinθ r=1+2sinθand r=−1+2sinθ r=3sinθand r=3sin(−θ) r=2cosθand r=2cos(−θ) r=1+3cosθand r=1−3cosθ
To see the Review answers, open this PDF file and look for section 6.8.