Find the Exact Value arcsin(sin(-(2pi)/3))

Problem

arcsin(sin(−(2*π)/3))

Solution

1. Evaluate the inner trigonometric function sin(−(2*π)/3)

2. Identify the reference angle for −(2*π)/3 which is π/3

3. Determine the value based on the unit circle. Since −(2*π)/3 is in Quadrant III, the sine value is negative.

sin(−(2*π)/3)=−√(,3)/2

4. Substitute this value back into the inverse sine function.

arcsin(−√(,3)/2)

5. Apply the range restriction for the inverse sine function, which is [−π/2,π/2]

6. Find the angle θ in the interval [−π/2,π/2] such that sin(θ)=−√(,3)/2

θ=−π/3

Final Answer

arcsin(sin(−(2*π)/3))=−π/3

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