The value of following expression is sin 31π/3?

To find the value of the expression sin(31π/3), we need to first simplify the angle.

Simplify the angle: The sine function has a period of 2π, meaning sin(θ) = sin(θ + 2nπ) for any integer n. So, we want to reduce 31π/3 to an equivalent angle between 0 and 2π.

First, divide 31π/3 by 2π to determine how many full periods (2π) fit into the angle: (31π/3) ÷ (2π) = 31/6.

This means the angle is 31/6 full periods of 2π, or 5 full periods with a remainder.

Find the remainder: To find the remainder, subtract the full periods (5 * 2π = 10π): 31π/3 - 10π = 31π/3 - 30π/3 = π/3.

So, 31π/3 is equivalent to π/3.

Evaluate the sine of π/3: From trigonometric values, we know that: sin(π/3) = √3/2.

Thus, the value of sin(31π/3) is √3/2.