Question

The graphs of r = cos theta and r = 2 cos 2 theta intersect in how many points?

Answer 1

I know one will be a rose and one will be a circle but I don't know how to calculate intersection

Answer 2

find the points of intersection.... by solving

alternatively... graph both curves

2cos(2x) takes pi for 1 period... while cos(x) takes 2pi for a period....

so they intersect twice over (0, pi) or 4 times over (0, 2pi)

Answer 3

This is an equation reducible to a quadratic

2cos(2x) = 2(cos^2x - sin^2(x)

using sin^2 x = 1 - cos^2(x)

then 2cos(2x) = 2(2cos^2(x) - 1)

the point of intersection is found by equating both sides

2(2cos^2(x) -1) = cos(x)

4cos^2(x) - cos(x) - 2 = 0

solve the quadratic using the general quadratic formula

cos(x) =\[(1 \pm \sqrt{32})/8\]

Answer 4

oops should be
\[\cos(x) = (1 \pm \sqrt{33})/8\]

Answer 5

I'll leave you to determine the domain of x