what is the value of sin pie/7 × sin 2pie/7 ×sin3pie/7 is

We need to find the value of the expression:

sin(π/7) × sin(2π/7) × sin(3π/7)

Step 1: Using a Known Identity
A well-known trigonometric identity states:

sin(π/n) × sin(2π/n) × ... × sin((n-1)π/n) = n / (2^(n-1))

For n = 7, this identity simplifies to:

sin(π/7) × sin(2π/7) × sin(3π/7) × sin(4π/7) × sin(5π/7) × sin(6π/7) = 7 / 2^6

Since we know that:

sin(π - x) = sin x

It follows that:

sin(4π/7) = sin(3π/7),
sin(5π/7) = sin(2π/7),
sin(6π/7) = sin(π/7)

Thus, rewriting the equation:

(sin(π/7) × sin(2π/7) × sin(3π/7))^2 = 7 / 64

Step 2: Taking Square Root
Taking the square root on both sides:

sin(π/7) × sin(2π/7) × sin(3π/7) = √(7/64)

Since √64 = 8, we get:

sin(π/7) × sin(2π/7) × sin(3π/7) = √7 / 8

Final Answer:
sin(π/7) × sin(2π/7) × sin(3π/7) = √7 / 8