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Question 62

$$16\sin(20°)\sin(40°)\sin(80°)$$ is equal to

We need to find the value of $$16\sin(20°)\sin(40°)\sin(80°)$$.

A well-known product-to-sum identity states that $$\sin\theta \cdot \sin(60° - \theta) \cdot \sin(60° + \theta) = \frac{\sin 3\theta}{4}$$.

Substituting θ = 20° into this identity gives $$\sin(20°) \cdot \sin(40°) \cdot \sin(80°) = \sin(20°) \cdot \sin(60° - 20°) \cdot \sin(60° + 20°)$$.

It follows that $$= \frac{\sin(60°)}{4} = \frac{\sqrt{3}/2}{4} = \frac{\sqrt{3}}{8}$$.

Multiplying both sides by 16 yields $$16\sin(20°)\sin(40°)\sin(80°) = 16 \times \frac{\sqrt{3}}{8} = 2\sqrt{3}$$.

The answer is Option B: $$2\sqrt{3}$$.

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