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How to Solve the Trigonometric Expression sin(π/14) × sin(3π/14) × sin(5π/14)
How to Solve the Trigonometric Expression sin(π/14) × sin(3π/14) × sin(5π/14)
Understanding and applying trigonometric identities is a fundamental skill that can simplify complex trigonometric expressions. In this article, we will explore a specific expression, sin(π/14) × sin(3π/14) × sin(5π/14), and demonstrate how to simplify it using trigonometric product identities.
Using a Specific Trigonometric Identity
The expression sin(π/14) × sin(3π/14) × sin(5π/14) can be simplified using a known trigonometric identity that relates the product of sines to a simpler expression. This identity for three sine products of the form sin(π/n) × sin(3π/n) × sin(5π/n) is given by:
sin(π/n) × sin(3π/n) × sin(5π/n) n/(2^{n-1})
For our specific case where n 14, the identity becomes:
sin(π/14) × sin(3π/14) × sin(5π/14) 14/(2^{14-1}) 14/8192
Further simplifying, we get:
14/8192 7/4096
Step-by-step Solution
Let's break down the steps involved in solving the expression:
First, let's denote π/14 as θ. Therefore, the expression becomes: sin(θ) × sin(3θ) × sin(5θ) Now, we will convert sin(3θ) into cos(4θ) and sin(5θ) into cos(2θ) using trigonometric transformation formulae.Trigonometric Transformation
Using the identity:
sin x cos(π/2 - x)
We can convert sin(3π/14) into cos(4π/14) and sin(5π/14) into cos(2π/14). Thus, the expression becomes:
sin(π/14) × cos(4π/14) × cos(2π/14)
Let's assume π/14 as x. So, the expression becomes:
sin x × cos 2x × cos 4x
Further Simplification
To simplify this further, we can multiply and divide by cos x and cos 2x successively:
2sin x cos x cos 2x cos 4x / 2 cos x
Which simplifies to:
2sin 2x cos 2x cos 4x / 4 cos x
And further to:
sin 4x cos 4x / 8 cos x
Which becomes:
sin 8x / 8 cos x
Finally, we have:
sin(8π/14) / 8 cos(π/14)
Since sin(8π/14) cos(π/14), we get:
1/8
This step-by-step approach helps in understanding the underlying trigonometric identities and simplifications.