find the exact value of arcsin(sin(8pi/3))
To find the exact value of arcsin(sin(8π/3)), we need to determine the angle θ that satisfies sin(θ) = sin(8π/3).
First, let's simplify 8π/3. Since 1 full rotation is equal to 2π, we can rewrite 8π/3 as (8π/3) - (2π) = (2π/3).
Now our equation becomes sin(θ) = sin(2π/3).
The sine function is periodic with a period of 2π. This means that sin(θ) = sin(2π/3) is also true for θ = (2π/3) + 2πk, where k is an integer.
Since we want the exact value of arcsin(sin(8π/3)), we need to find the unique angle in the range [-π/2, π/2] that satisfies sin(θ) = sin(2π/3). This will be the principal value of arcsin(sin(8π/3)).
To find this angle, we can consider the unit circle. The reference angle for (2π/3) lies in the second quadrant.
In the second quadrant, sine is positive. Therefore, sin(2π/3) = sin(π - (2π/3)).
Now, we can see that π - (2π/3) = π/3.
So, the unique angle in the range [-π/2, π/2] that satisfies sin(θ) = sin(2π/3) is θ = π/3.
Therefore, the exact value of arcsin(sin(8π/3)) is π/3.