Find the Exact Value arcsin(sin((4pi)/9))

Problem

arcsin(sin((4*π)/9))

Solution

1. Identify the range of the principal inverse sine function, which is [−π/2,π/2]

2. Check if the input angle θ=(4*π)/9 falls within this interval.

3. Compare the values: since −π/2=−(4.5*π)/9 and π/2=(4.5*π)/9 the angle (4*π)/9 is within the interval [−(4.5*π)/9,(4.5*π)/9]

4. Apply the property arcsin(sin(x))=x for x∈[−π/2,π/2]

Final Answer

arcsin(sin((4*π)/9))=(4*π)/9

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