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NDA 2025 Maths · Mathematics · Trigonometric Functions and Identities · hard

The value of sin 20^°sin 40^°sin 60^°sin 80^° is

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The value of sin⁡20∘sin⁡40∘sin⁡60∘sin⁡80∘\sin20^\circ\sin40^\circ\sin60^\circ\sin80^\circ is
  1. A. 116\dfrac1{16}
  2. B. 38\dfrac38
  3. C. 316\dfrac{3}{16}
  4. D. 18\dfrac18
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Answer: (C) 316\dfrac{3}{16}

Use sin⁡θsin⁡(60∘−θ)sin⁡(60∘+θ)=14sin⁡3θ\sin\theta\sin(60^\circ-\theta)\sin(60^\circ+\theta)=\frac14\sin3\theta with θ=20∘\theta=20^\circ: sin⁡20∘sin⁡40∘sin⁡80∘=14sin⁡60∘=38\sin20^\circ\sin40^\circ\sin80^\circ=\frac14\sin60^\circ=\frac{\sqrt3}{8}. Multiplying by sin⁡60∘=32\sin60^\circ=\frac{\sqrt3}2 gives 316\frac3{16}.
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