find the exact value of the expression.\n(\tan left(sin ^{-1} \frac{5}{6}\right))\nselect the correct choice…

find the exact value of the expression.\n(\tan left(sin ^{-1} \frac{5}{6}\right))\nselect the correct choice and fill in any answer boxes in your choice below.\na. (\tan left(sin ^{-1} \frac{5}{6}\right)=)\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)\nb. there is no solution.

find the exact value of the expression.\n(\tan left(sin ^{-1} \frac{5}{6}\right))\nselect the correct choice and fill in any answer boxes in your choice below.\na. (\tan left(sin ^{-1} \frac{5}{6}\right)=)\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)\nb. there is no solution.

Answer

Explanation:

Step1: Let (\theta=\sin^{-1}\frac{5}{6})

By the definition of the inverse - sine function, (\sin\theta=\frac{5}{6}), where (-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}). Since (\sin\theta=\frac{5}{6}>0), then (0 < \theta<\frac{\pi}{2}). Using the Pythagorean identity (\sin^{2}\theta+\cos^{2}\theta = 1), we can solve for (\cos\theta): [ \begin{align*} \cos\theta&=\sqrt{1-\sin^{2}\theta}\ &=\sqrt{1 - (\frac{5}{6})^{2}}\ &=\sqrt{1-\frac{25}{36}}\ &=\sqrt{\frac{36 - 25}{36}}\ &=\sqrt{\frac{11}{36}}\ &=\frac{\sqrt{11}}{6} \end{align*} ]

Step2: Use the formula for (\tan\theta)

We know that (\tan\theta=\frac{\sin\theta}{\cos\theta}). Since (\sin\theta=\frac{5}{6}) and (\cos\theta=\frac{\sqrt{11}}{6}), then (\tan\theta=\frac{\frac{5}{6}}{\frac{\sqrt{11}}{6}}). [ \tan\theta=\frac{5}{\sqrt{11}}=\frac{5\sqrt{11}}{11} ]

Answer:

A. (\tan\left(\sin^{-1}\frac{5}{6}\right)=\frac{5\sqrt{11}}{11})