Find the exact value of the slope of the line which is tangent to the curve given by the equation r = 2 + cos θ at θ=pi/2. You must show your work.

Question

My answer is "-1" but my friend has "1/2". Which answer is correct?

Answer 1

r = 2+cosθ
r' = -sinθ
r'(pi/2) = -1

The slope of the line is dy/dx
y = rsinθ = 2r + cosθsinθ = 2r + 1/2 sin2θ
dy/dθ = 2r' + cos2θ
dy/dθ at pi/2 = -1

x = rcosθ = 2r + cos^2θ
dx/dθ = 2r' + 2cosθ(-sinθ)
dx/dθ at pi/2 = -2

so, dy/dx = (dy/dθ)/(dx/dθ) at pi/2 = 1/2

Maybe next time you could show your work. Or else ...