Question
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?
Answers
Reiny
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°).
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°).