Part 2/5:
This formulation indicates that the ratio of the opposite side to the hypotenuse in a right triangle is ( \frac{1}{2} ). From the unit circle or known values, we know that this relationship holds true at an angle of ( 30^\circ ) or ( \frac{\pi}{6} ) radians. Thus, we conclude:
[
\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}
]
To further substantiate this, if we were to employ the Pythagorean theorem, where we have a triangle with sides ( 1 ) and ( 2 ), the length of the adjacent side would be determined as follows:
[
\text{Adjacent} = \sqrt{2^2 - 1^2} = \sqrt{3}
]
Thus, the angle ( \theta ) corresponding to this setup indeed verifies our finding.