Let α, β, γ and δ be the four least positive angles, arranged in increasing order, whose sine is the positive value x. Then the expression 4 sin(α/2) + 3 sin(β/2) + 2 sin(γ/2) + sin(δ/2) equals

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2√(1+x). The four least positive angles with sine x are alpha, pi-alpha, 2pi+alpha, 3pi-alpha (increasing). Substituting their half-angles into 4 sin(a/2)+3 sin(b/2)+2 sin(g/2)+sin(d/2) and simplifying gives 2 sqrt(1+x), confirmed numerically for several x.

Let α, β, γ and δ be the four least positive angles, arranged in increasing order, whose sine is the positive value x. Then the expression 4 sin(α/2) + 3 sin(β/2) + 2 sin(γ/2) + sin(δ/2) equals

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