For every angle θ, the sum sin(θ − 60°) + sinθ + sin(θ + 60°) is equal to:

The outer terms combine by the sum-to-product rule: sin(θ − 60°) + sin(θ + 60°) = 2·sinθ·cos60° = sinθ. Adding the middle term gives 2sinθ. Check at θ = 30°: −0.5 + 0.5 + 1 = 1 = 2sin30°. Three angles in arithmetic progression cancel to 0 only when the spacing is 120°; with a 60° spacing the outer pair reinforces the middle term instead, and treating all three sines as equal gives 3sinθ.