Let α, β, γ be real numbers with sin α, sin β, sin γ all nonzero. Define
Δ = | sin²α sinα cosα cos²α |
| sin²β sinβ cosβ cos²β |
| sin²γ sinγ cosγ cos²γ |.
What is the greatest value Δ can attain?

Question

Accepted Answer

None of these. Using sin²=(1-cos2θ)/2, sin·cos=sin2θ/2, cos²=(1+cos2θ)/2, the three columns are independent combinations of 1, sin2θ, cos2θ, so Δ is generally nonzero. Its maximum value is about 0.6495 (=3√3/8), which is none of 1, 0, or -1/2; hence None of these.

Let α, β, γ be real numbers with sin α, sin β, sin γ all nonzero. Define Δ = | sin²α sinα cosα cos²α | | sin²β sinβ cosβ cos²β | | sin²γ sinγ cosγ cos²γ |. What is the greatest value Δ can attain?

Solution

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