What is the <strong>MAXIMUM</strong> value of 8 \sin \theta - 4 \sin^2 \theta?

Explanation

Let x = \sin \theta. The expression is a quadratic function f(x) = -4x^2 + 8x. The maximum value for a parabola opening downwards occurs at x = -\frac{b}{2a} = -\frac{8}{2(-4)} = 1. Since x = \sin \theta \leq 1, this is a valid maximum. Substituting x=1 yields -4(1)^2 + 8(1) = 4.

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