Prove that: [tex]$\sin 20^{\circ} \cdot \sin 30^{\circ} \cdot \sin 40^{\circ} \cdot \sin 80^{\circ}=\frac{\sqrt{3}}{16}$[/tex].
Answer (2)
Recognize that sin 3 0 ∘ = 2 1 and substitute it into the expression.
Apply the trigonometric identity sin x ⋅ sin ( 6 0 ∘ − x ) ⋅ sin ( 6 0 ∘ + x ) = 4 1 sin 3 x with x = 2 0 ∘ .
Simplify the expression using sin 6 0 ∘ = 2 3 .
Conclude that sin 2 0 ∘ ⋅ sin 3 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 16 3 .
16 3
Explanation
Problem Analysis and Initial Setup We are asked to prove the trigonometric identity: sin 2 0 ∘ ⋅ sin 3 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 16 3 We will start by simplifying the left-hand side of the equation.
Using the value of sin 30 degrees We know that sin 3 0 ∘ = 2 1 . Substituting this value into the left-hand side, we get: sin 2 0 ∘ ⋅ 2 1 ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 2 1 ⋅ sin 2 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ Now, we need to simplify the product sin 2 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ .
Applying the Trigonometric Identity We can use the trigonometric identity: sin x ⋅ sin ( 6 0 ∘ − x ) ⋅ sin ( 6 0 ∘ + x ) = 4 1 sin 3 x Let x = 2 0 ∘ . Then we have: sin 2 0 ∘ ⋅ sin ( 6 0 ∘ − 2 0 ∘ ) ⋅ sin ( 6 0 ∘ + 2 0 ∘ ) = sin 2 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ Using the identity, we get: sin 2 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 4 1 sin ( 3 ⋅ 2 0 ∘ ) = 4 1 sin 6 0 ∘ Since sin 6 0 ∘ = 2 3 , we have: sin 2 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 4 1 ⋅ 2 3 = 8 3
Final Substitution and Conclusion Now, substitute this result back into the expression we had after using sin 3 0 ∘ = 2 1 :
2 1 ⋅ sin 2 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 2 1 ⋅ 8 3 = 16 3 Thus, we have shown that: sin 2 0 ∘ ⋅ sin 3 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 16 3
Examples
Trigonometric identities are useful in various fields such as physics, engineering, and computer graphics. For example, in signal processing, trigonometric functions are used to analyze and manipulate signals. Understanding and proving trigonometric identities helps in simplifying complex expressions and solving problems related to wave phenomena, oscillations, and harmonic motion. These identities also play a crucial role in simplifying calculations in fields like surveying and navigation, where angles and distances are frequently involved.
We proved that sin 2 0 ∘ ⋅ sin 3 0 ∘ ⋅ sin 4 0 ∘ ⋅ sin 8 0 ∘ = 16 3 by substituting the known value of sin 3 0 ∘ and applying a trigonometric identity. Our calculations were simplified through the use of trigonometric identities and substitution, leading us to the conclusion. The identity is verified as true.
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