Find the exact value of the slope of the line which is tangent to the curve given by the equation

Question

Find the exact value of the slope of the line which is tangent to the curve given by the equation

r = 1 + cos θ at  . You must show your work.

Answer 1

I believe the way to solve this is to set up the Cartesian equations

[y=sinθ(1+cosθ) and x=sinθ(1+cosθ)].

Then differentiate each equation with respect to θ, i.e. get dx/dθ and dy/dθ.

Then divide the two expressions to get dy/dx which will given you an expression in θθ

Evaluate at θ=π/2.

This will given you the slope = 1,

If you graph the polar equation you will get a cardioid and the slope looks correct.

I can't remember why this is correct but I believe it is.