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

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

  • calculus -

    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

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

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