Question

Find the exact value of the given expression. $$ \cos \left(\sin ^{-1} \frac{1}{2}\right) $$

Short answer

The exact value of the given expression is \(\frac{\sqrt{3}}{2}\).

Step-by-step solution

Evaluate the inverse sine function

First, we will evaluate the inner expression, which is the inverse sine function acting on 1/2. To do this, we recall that: \[ \sin^{-1}(x) = \theta \Rightarrow \sin(\theta) = x \] Now, substituting x with 1/2, we look for the angle \(\theta\) such that \(\sin(\theta) = \frac{1}{2}\) From our knowledge of known trigonometric values or by examining the sine function in the unit circle, we know that: \[ \sin \frac{\pi}{6} = \frac{1}{2} \] So, now we have the angle, \(\theta = \frac{\pi}{6}\).

Calculate the cosine value of the angle

Now that we have the angle, we can compute the cosine of that angle. We have: \[ \cos(\theta) = \cos\left(\frac{\pi}{6}\right) \] Recall that we have a list of angles and their corresponding cosine values from the unit circle. In this case, we know that: \[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

State the final answer

Putting everything together, we have the exact value of the given expression: \[ \cos\left(\sin^{-1}\frac{1}{2}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]