Concept:The principal value of sin−1x is the unique angle θ in the interval [−2π,2π] such that sinθ=x.Explanation:We need to find sin−1(−23). Recognize that sin(−3π)=−23. Since −3π lies in the range [−2π,2π], this is the principal value. Therefore, sin−1(−23)=−3π.A helpful table of principal value ranges for inverse trigonometric functions:
Function
Domain
Range of Principal Value
sin-1 x
[-1, 1]
[-π/2, π/2]
cos-1 x
[-1, 1]
[0, π]
csc-1 x
R - (-1, 1)
[-π/2, π/2] - {0}
sec-1 x
R - (-1, 1)
[0, π] - {π/2}
tan-1 x
R
(-π/2, π/2)
cot-1 x
R
(0, π)
Answer:The principal value is −3π, which corresponds to option B.