find the exact value of the real number y.\n\n$y = \\sin^{-1}\\left(\\frac{1}{2}\\right)$\n\na…

find the exact value of the real number y.\n\n$y = \\sin^{-1}\\left(\\frac{1}{2}\\right)$\n\na. $\\frac{\\pi}{6}$\nb. $-\\frac{\\pi}{6}$\nc. 0\nd. $\\frac{\\pi}{4}$

find the exact value of the real number y.\n\n$y = \\sin^{-1}\\left(\\frac{1}{2}\\right)$\n\na. $\\frac{\\pi}{6}$\nb. $-\\frac{\\pi}{6}$\nc. 0\nd. $\\frac{\\pi}{4}$

Answer

Explanation:

Step1: Recall the definition of inverse sine function

The inverse sine function (y = \sin^{-1}(x)) has a range ([-\frac{\pi}{2},\frac{\pi}{2}]) and (\sin^{-1}(\sin\theta)=\theta) when (\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]). We need to find (\theta) such that (\sin\theta=\frac{1}{2}) and (\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]).

Step2: Use the unit - circle or special angles

We know that (\sin\frac{\pi}{6}=\frac{1}{2}) and (\frac{\pi}{6}\in[-\frac{\pi}{2},\frac{\pi}{2}]).

Answer:

A. (\frac{\pi}{6})