For what values of theta, on the polar curve r=theta, 0<=theta<=2pi, are there horizontal tangent curves? Vertical?

Question

I got one answer for the horizontal tangent. and it's theta = 0 but what are the other ones?/

Answer 1

since r = θ

x = r cosθ
y = r sinθ

dy/dθ = r' sinθ + r cosθ = sinθ + θcosθ
dx/dθ = r' cosθ - r sinθ = cosθ - θsinθ

dy/dx = (sinθ + θcosθ)/(cosθ - θsinθ)

You are correct that dy/dx = 0 when θ=0
It is also zero when x = 2.04 and 4.88
(approximately)

The tangent is vertical when the denominator is zero: x = 0.84 and 3.45