Prove that: [tex]8 \cos 10^{\circ} \cdot \cos 50^{\circ} \cdot \cos 70^{\circ}=\sqrt{3}[/tex].
Answer (2)
Use the product-to-sum identity to simplify 8 cos 1 0 ∘ cos 5 0 ∘ cos 7 0 ∘ .
Apply the identity 2 cos A cos B = cos ( A + B ) + cos ( A − B ) to cos 1 0 ∘ cos 5 0 ∘ and then to cos 4 0 ∘ cos 7 0 ∘ .
Use the fact that cos ( 18 0 ∘ − x ) = − cos x to simplify the expression further.
Simplify to get 2 cos 3 0 ∘ = 3 , thus 8 cos 1 0 ∘ cos 5 0 ∘ cos 7 0 ∘ = 3 .
3
Explanation
Problem Analysis We are tasked to prove the trigonometric identity 8 cos 1 0 ∘ ⋅ cos 5 0 ∘ ⋅ cos 7 0 ∘ = 3 . To do this, we will use trigonometric identities to simplify the left-hand side of the equation until it equals the right-hand side.
Applying Product-to-Sum Identity We will use the product-to-sum identity 2 cos A cos B = cos ( A + B ) + cos ( A − B ) to simplify the expression. First, let's multiply cos 1 0 ∘ and cos 5 0 ∘ :
2 cos 1 0 ∘ cos 5 0 ∘ = cos ( 1 0 ∘ + 5 0 ∘ ) + cos ( 5 0 ∘ − 1 0 ∘ ) = cos 6 0 ∘ + cos 4 0 ∘
Substitution Now, substitute this result back into the original expression:
8 cos 1 0 ∘ cos 5 0 ∘ cos 7 0 ∘ = 4 ( cos 6 0 ∘ + cos 4 0 ∘ ) cos 7 0 ∘ = 4 ( 2 1 + cos 4 0 ∘ ) cos 7 0 ∘ = 2 cos 7 0 ∘ + 4 cos 4 0 ∘ cos 7 0 ∘
Applying Product-to-Sum Identity Again Next, multiply cos 4 0 ∘ and cos 7 0 ∘ using the product-to-sum identity:
2 cos 4 0 ∘ cos 7 0 ∘ = cos ( 4 0 ∘ + 7 0 ∘ ) + cos ( 7 0 ∘ − 4 0 ∘ ) = cos 11 0 ∘ + cos 3 0 ∘
Substitution Substitute this result back into the expression:
2 cos 7 0 ∘ + 4 cos 4 0 ∘ cos 7 0 ∘ = 2 cos 7 0 ∘ + 2 ( cos 11 0 ∘ + cos 3 0 ∘ )
Using the Property of Cosine Now, simplify using the fact that cos ( 18 0 ∘ − x ) = − cos x , so cos 11 0 ∘ = cos ( 18 0 ∘ − 7 0 ∘ ) = − cos 7 0 ∘ .
Simplification Substitute this back into the expression:
2 cos 7 0 ∘ + 2 ( − cos 7 0 ∘ + cos 3 0 ∘ ) = 2 cos 7 0 ∘ − 2 cos 7 0 ∘ + 2 cos 3 0 ∘ = 2 cos 3 0 ∘
Final Simplification Since cos 3 0 ∘ = 2 3 , the expression simplifies to:
2 ⋅ 2 3 = 3
Conclusion Therefore, we have shown that 8 cos 1 0 ∘ ⋅ cos 5 0 ∘ ⋅ cos 7 0 ∘ = 3 .
Examples
Trigonometric identities are fundamental in various fields such as physics, engineering, and computer graphics. For instance, in signal processing, these identities are used to analyze and manipulate waveforms. In computer graphics, they help in calculating angles and distances for rendering 3D objects. Moreover, in physics, they are essential for analyzing oscillatory motions and wave phenomena. Understanding and applying trigonometric identities allows us to simplify complex expressions and solve problems in these diverse areas.
Using trigonometric identities, we simplify and show that 8 cos 1 0 ∘ ⋅ cos 5 0 ∘ ⋅ cos 7 0 ∘ = 3 by applying the product-to-sum identity and using known values of cosine angles.
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