The sum sinθ + sin(180° − θ) + sin(180° + θ) + sin(360° − θ) equals:

A supplement leaves the sine alone, so sin(180° − θ) = +sinθ, while sin(180° + θ) = −sinθ and sin(360° − θ) = −sinθ. The four terms are sinθ + sinθ − sinθ − sinθ = 0. Check at θ = 40°: 0.643 + 0.643 − 0.643 − 0.643 = 0. Treating every allied angle as sign-free gives 4sinθ, and catching only the third flip leaves 2sinθ.