Find the polar equation r=8cos3theta, find the maximum value of lrl and any zeros of r. Verify your answers numerically.
|cos3θ| <= 1
cos3θ=0 when 3θ is an odd multiple of π/2
what is the next step to answering the question?
Steve had done the above steps for you, see the first of the Related Questions for you.
First, it might help if you looked at the graph
http://www.wolframalpha.com/input/?i=polar+plot+r%3D8cos(3%C3%98)
Steve told you:
|cos 3Ø| ? 1
so |8cos 3Ø| ? 8
| r | ? 8 , so that's done!
for the zeros's of r, Steve said:
8cos 3Ø = 0
cos 3Ø = 0
3Ø = ?/2 or 3Ø = 3?/2 <---- you should know this
Ø = ?/6 or Ø = ?/2
but the period of cos 3Ø is 2?/3 ,
so adding/subtracting 2?/3 to any of the answers already found will give more solutions.
starting with Ø= ?/2
e.g. ?/2 + 2?/3 = 7?/6
7?/6 + 2?/3 = 11?/6 <--- last one before 2?
Starting with Ø = ?/6
?/6 + 2?/3 = 5?/6
5?/6 + 2?/3 = 3?/2
3?/2 + 2?/3 = 13?/6 , which is > 2? by ?/6 , so we are repeating.
So for just one rotations, we have
Ø = ?/6, 5?/6 , 3?/2, ?/2, 7?/6, and 11?/6
If you find it easier to think in degrees
Ø = 30°, 150° ,270°, 90°, 210°, and 330°
checking one of these, e.g. Ø = 210°
r = 8cos(3(210°))
= 8 cos 630°
= 8(0) = 0
Notice that when you sketch tangents to the curve, you can do this at 30° (same as 210° line), 150° (same as 330° line) and at 90° (same as the 270°).
The next step to answering the question is to find the maximum value of |r| and any zeros of r.
The next step to answering the question would be to find the maximum value of |r| and determine the zeros of r.
To find the maximum value of |r|, we can substitute the values of θ that make cos3θ equal to 1 or -1 into the polar equation r = 8cos3θ and calculate the corresponding values of r.
When cos3θ = 1, we have 3θ = 2nπ, where n is an integer. Solving for θ, we get θ = (2nπ)/3.
When cos3θ = -1, we have 3θ = (2n+1)π, where n is an integer. Solving for θ, we get θ = ((2n+1)π)/3.
Now we substitute these values of θ back into the polar equation r = 8cos3θ to get the corresponding values of r.
For θ = (2nπ)/3:
r = 8cos3((2nπ)/3) = 8cos(2nπ) = 8(1) = 8
For θ = ((2n+1)π)/3:
r = 8cos3(((2n+1)π)/3) = 8cos(2nπ + π) = 8(-1) = -8
Hence, the maximum value of |r| is 8.
To determine the zeros of r, we need to find the values of θ that make r equal to zero. In other words, we need to find the values of θ that satisfy the equation r = 0.
Setting r = 8cos3θ to zero, we have:
8cos3θ = 0
For cos3θ = 0, we know that 3θ is an odd multiple of π/2. Hence, we can solve for θ:
3θ = (2n+1)π/2, where n is an integer.
θ = (2n+1)π/6, where n is an integer.
So, the zeros of r occur at θ = (2n+1)π/6, where n is an integer.
To verify the answers numerically, you can substitute different values of θ into the polar equation r = 8cos3θ and calculate the corresponding values of r. Verify that the maximum value of |r| is 8, and check that the values of r are indeed zero at the zeros of r that we found.