You have polar functions r1=3cos(2?) and r2=3cos(t) 1. What are the intersection

Question

You have polar functions r1=3cos(2?) and r2=3cos(t)

1. What are the intersection points of these curves? (Show work.)
2. Find the outside r1 and inside r2. Explain how you arrived at the limits of integration.
3. Find the length of the curve r1.

Explanation / Answer

r_1 = r_2 3 cos(2?) = 3cos(?) cos(2?) = cos(?) ---> knowing cos(2?) = 1 - 2sin^2(?) 1 - 2sin^2(?) = cos(?) ---> square both sides 1 - 4sin^2(?) + 4sin^4(?) = cos^2(?) 1 - 4sin^2(?) + 4sin^4(?) = 1 - sin^2(?) 1 - 4sin^2(?) + 4sin^4(?) - 1 + sin^2(?) = 0 - 3sin^2(?) + 4sin^4(?) = 0 4sin^4(?) - 3sin^2(?) = 0 sin^2(?) * ( 4sin^2(?) - 3 ) = 0 sin^2(?) = 0 ---> sin(?) = 0 ---> ? = 0, p, 2p, .... or pn where n is an integer. 4sin^2(?) - 3 = 0 ---> 4sin^2(?) = 3 ---> sin(?) = +/- v(3)/2 ---> ? = p/3, 2p/3, 4p/3, 5p/3, --> p/3 + pn & 2p/3 + pn where n is an integer.

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