SOLUTION: Please help me with this Calculus II question: Find all points of intersection of the given curves. (Assume 0 ≤ θ ≤ π. Order your answers from smallest

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Question 809616: Please help me with this Calculus II question:
Find all points of intersection of the given curves. (Assume
0 ≤ θ ≤ π.
Order your answers from smallest to largest θ. If an intersection occurs at the pole, enter POLE in the first answer blank.)

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Please help me with this Calculus II question:
Find all points of intersection of the given curves. (Assume
0 ≤ θ ≤ π.
Order your answers from smallest to largest θ. If an intersection occurs at the pole, enter POLE in the first answer blank.)
Here are the curves:
r = cos 3θ, r = sin 3θ
cos 3θ = sin 3θ since both = r 

 Divide both sides by cos 3θ

cos%283theta%29%2Fcos%283theta%29%22%22=%22%22sin%283theta%29%2Fcos%283theta%29

 
1 = tan(3θ)

So 3θ has referent angle 45° or pi%2F4, The tangent is positive in 
QI and QIII 

Since we are requiring that  0 ≤ θ ≤ π, that means that
                             0 ≤ 3θ ≤ 3π

So 3θ can be pi%2F4, 5pi%2F4, and 9pi%2F4

So θ can be pi%2F12, 5pi%2F12, and 9pi%2F12, which reduces to  3pi%2F4

We find the value of r for each of those three values for θ, by substituting
in either equation for r

when θ = pi%2F12,r = cos 3θ = cos%283pi%2F12%29 = cos%28pi%2F4%29 = sqrt%282%29%2F2

So the first point is (r,θ) = %28matrix%281%2C3%2C++++sqrt%282%29%2F2%2C%22%2C%22%2Cpi%2F12%29%29 

when θ = 5pi%2F12,r = cos 3θ = cos%2815pi%2F12%29 = cos%285pi%2F4%29 = sqrt%282%29%2F2

So the second point is (r,θ) = %28matrix%281%2C3%2C++++sqrt%282%29%2F2%2C%22%2C%22%2C5pi%2F12%29%29

when θ = pi%2F12,r = cos 3θ = cos%283pi%2F4%29 = sqrt%282%29%2F2

So the third point is (r,θ) = %28matrix%281%2C3%2C++++sqrt%282%29%2F2%2C%22%2C%22%2C3pi%2F4%29%29

Edwin