Find the Exact Value arcsin(sin((5pi)/4))

Problem

arcsin(sin((5*π)/4))

Solution

1. Evaluate the inner sine function. The angle (5*π)/4 is in the third quadrant, where the sine value is negative.

sin((5*π)/4)=−√(,2)/2

2. Identify the range of the inverse sine function. The output of arcsin(x) must fall within the interval [−π/2,π/2]

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

θ=−π/4

4. Conclude that since −π/4 is the principal value within the required range, it is the exact value of the expression.

Final Answer

arcsin(sin((5*π)/4))=−π/4

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