Simplify arcsin(- square root of 3/2)

Problem

arcsin(−√(,3)/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(−√(,3)/2)=−arcsin(√(,3)/2)

4. Determine the angle in the interval [0,π/2] whose sine value is √(,3)/2

5. Recall from the unit circle that sin(π/3)=√(,3)/2

6. Substitute the value back into the expression:

−arcsin(√(,3)/2)=−π/3

Final Answer

arcsin(−√(,3)/2)=−π/3

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