Question

find the slope of the line tangent to the polar curves r=4cos2θ at the tip of the leaves

Answer 1

\(x= rcos \theta\rightarrow x= 4cos(2\theta) cos\theta\)

\(y = r sin\theta\rightarrow y=4cos(2\theta sin\theta\)

the slope of the curve at any point is

\[\dfrac{dy}{dx}=\dfrac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\]

Answer 2

Your polar function has tip of the leaves (4,0), (0,4),(-4,0), and (0,-4)

At those points, \(\theta = 0, \pi/2, \pi,3\pi/2\)
replace to get the result