Simplify arcsin(- square root of 2/2)

Problem

arcsin(−√(,2)/2)

Solution

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

2. Apply the property of odd functions for inverse sine, which states arcsin(−x)=−arcsin(x)

3. Rewrite the expression using this property:

arcsin(−√(,2)/2)=−arcsin(√(,2)/2)

4. Determine the angle in the interval [0,π/2] whose sine value is √(,2)/2

5. Recall from the unit circle that sin(π/4)=√(,2)/2

6. Substitute this value back into the expression:

−arcsin(√(,2)/2)=−π/4

Final Answer

arcsin(−√(,2)/2)=−π/4

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