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

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

find the exact value of the expression. \n\n \\( \\cos \\left \\sin ^ { - 1 } \\left( - \\frac { \\sqrt { 3 } } { 2 } \\right) \\right \\) \n\n select the correct choice and fill in any answer boxes in your choice below. \n\n \\( \\bigcirc \\mathrm { a } \\) \n\n \\( \\cos \\left \\sin ^ { - 1 } \\left( - \\frac { \\sqrt { 3 } } { 2 } \\right) \\right = \\) \n\n (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) \n\n \\( \\bigcirc \\mathrm { b } \\). 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) for which (\sin\theta =-\frac{\sqrt{3}}{2}) and (\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]) is (\theta =-\frac{\pi}{3}).

Step2: Find (\cos\theta)

Since (\theta =-\frac{\pi}{3}), then (\cos\left[\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\cos\left(-\frac{\pi}{3}\right)). Using the property (\cos(-x)=\cos x), we have (\cos\left(-\frac{\pi}{3}\right)=\cos\frac{\pi}{3}). And we know that (\cos\frac{\pi}{3}=\frac{1}{2}).

Answer:

(\cos\left[\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\frac{1}{2}), so the answer is A. (\frac{1}{2})