Find the exact value of arcsinsin4π3.Write your answer in radians in terms of π.

To find the exact value of arcsin(sin(4π/3)), we first need to understand what the arcsin function does. The arcsin function, also known as the inverse sine function, undoes the operation of the sine function. In other words, if y = sin(x), then x = arcsin(y).

However, the arcsin function only returns values in the range -π/2 to π/2. This is because the sine function repeats every 2π, but we want the arcsin function to give a unique output. So, if we have a value of x that is outside the range -π/2 to π/2, we need to find an equivalent angle that is within this range.

The angle 4π/3 is equivalent to -2π/3 when reduced to the range -π/2 to π/2. This is because 4π/3 - 2π = -2π/3.

Therefore, arcsin(sin(4π/3)) = arcsin(sin(-2π/3)) = -2π/3.