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Chapter 0: Problem 48

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

Short Answer

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

Step by step solution

01

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, \(\theta = \frac{\pi}{6}\).
02

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: \[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
03

State the final answer

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

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