How many values of θ with 0° ≤ θ < 360° satisfy 2sinθ = tanθ?

Write tanθ = sinθ/cosθ (so cosθ ≠ 0) and multiply through: 2sinθ·cosθ = sinθ, i.e. sinθ(2cosθ − 1) = 0. From sinθ = 0 come θ = 0° and 180°; from cosθ = 1/2 come θ = 60° and 300°. That is 4 values. Dividing both sides by sinθ at the start silently throws away the first two.