find the exact value of the expression.\n sec left sin ^ { - 1 } left( - \frac { sqrt { 3 } } { 2 } \right)…

find the exact value of the expression.\n sec left sin ^ { - 1 } left( - \frac { sqrt { 3 } } { 2 } \right) \right \nselect the correct choice and fill in any answer boxes in your choice below.\noa \n sec left sin ^ { - 1 } left( - \frac { sqrt { 3 } } { 2 } \right) \right = \n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)\nob. there is no solution.

find the exact value of the expression.\n sec left sin ^ { - 1 } left( - \frac { sqrt { 3 } } { 2 } \right) \right \nselect the correct choice and fill in any answer boxes in your choice below.\noa \n sec left sin ^ { - 1 } left( - \frac { sqrt { 3 } } { 2 } \right) \right = \n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)\nob. there is no solution.

Answer

Explanation:

Step1: Let (\theta=\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right))

By the definition of the inverse - sine function, (\sin\theta =-\frac{\sqrt{3}}{2}) and (\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]). The value of (\theta) that satisfies (\sin\theta =-\frac{\sqrt{3}}{2}) in the interval (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]) is (\theta =-\frac{\pi}{3}).

Step2: Find (\sec\theta)

Since (\sec\theta=\frac{1}{\cos\theta}) and (\theta =-\frac{\pi}{3}), and (\cos(-\frac{\pi}{3})=\cos\frac{\pi}{3}=\frac{1}{2}) (because (\cos(-x)=\cos x)). Then (\sec\left(-\frac{\pi}{3}\right)=\frac{1}{\cos\left(-\frac{\pi}{3}\right)}).

Answer:

(\sec\left[\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=2)