Question

Find the exact value of the given expression. $$ \sin ^{-1} \frac{1}{2} $$

Short answer

The exact value of the given expression is \(\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}\).

Step-by-step solution

Understand the inverse sine function

The inverse sine function, written as \(\sin^{-1}\), is a function that takes a number between -1 and 1 and gives an angle in radians which is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). In other words, it is the function that "undoes" the sine function. So, given a value, it returns the angle in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) whose sine is equal to that value.

Identify the value we need to evaluate

In the given expression, we want to find the angle whose sine is \(\frac{1}{2}\): \[ \sin^{-1}\left(\frac{1}{2}\right) \]

Determine the angle with sine equal to \(\frac{1}{2}\)

We need to find an angle \(\alpha\) in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) such that: \[ \sin (\alpha) = \frac{1}{2} \] Using the properties of right triangles, we can find the angle by recalling the sine of specific angles in the unit circle. We know that \(sin(\frac{\pi}{6}) = \frac{1}{2}\). Therefore, the required angle is: \[ \alpha = \frac{\pi}{6} \]

Write the final answer

The exact value of the given expression is: \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]