SOLUTION: a. The equation r2 = 4cos2θ is of the form r^2 = a^2*cos2θ, so the graph is a lemniscate. What is the length of each loop? b. At what values of θ do the endpoints

Algebra ->  Trigonometry-basics -> SOLUTION: a. The equation r2 = 4cos2θ is of the form r^2 = a^2*cos2θ, so the graph is a lemniscate. What is the length of each loop? b. At what values of θ do the endpoints      Log On


   

Question 1103012: a. The equation r2 = 4cos2θ is of the form r^2 = a^2*cos2θ, so the graph is a lemniscate. What is the length of each loop?
b. At what values of θ do the endpoints of each loop occur?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
For a lemniscate of the form,
r%5E2=a%5E2%2Acos%282%2Atheta%29
the arc length is,
S=%28a%2Fsqrt%282pi%29%29%2AGAMMA%281%2F4%29%5E2 where GAMMA is the gamma function.
.
.
.
GAMMA%281%2F2%29=sqrt%28pi%29
So,
s=%282%2Fsqrt%282pi%29%29pi
s=sqrt%282pi%29
.
.
.
The maximum absolute value of the cosine is equal to 1 and occurs when the argument 2%2Atheta=0 and 2%2Atheta=pi. So then when,
theta=0
theta=%281%2F2%29pi
theta=pi
theta=%283%2F2%29pi
and equals zero when,
theta=%281%2F4%29pi
theta=%283%2F4%29pi
theta=%285%2F4%29pi
theta=%287%2F4%29pi