TechTorch

Location:HOME > Technology > content

Technology

How to Solve the Trigonometric Expression sin(π/14) × sin(3π/14) × sin(5π/14)

April 18, 2025Technology4843
How to Solve the Trigonometric Expression sin(π/14) × sin(3π/14) × sin

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.