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Chapter 5: Problem 5

Evaluate \(\cos \left(\cos ^{-1} \frac{1}{4}\right)\)

Short Answer

Expert verified
The expression \(\cos(\cos^{-1} \frac{1}{4})\) asks us to find the cosine of the angle whose cosine is \(\frac{1}{4}\). Since \(\cos(\theta) = \frac{1}{4}\), the expression simplifies to \(\frac{1}{4}\).

Step by step solution

01

Understand the Inverse Cosine Function

The inverse cosine function, written as \(\cos^{-1}\), is used to find the angle whose cosine is a given value. In our case, the value is \(\frac{1}{4}\), so we need to find an angle \(\theta\) such that \(\cos(\theta) = \frac{1}{4}\).
02

Determine the Angle

Let \(\theta = \cos^{-1}\frac{1}{4}\). By definition, this means that \(\cos(\theta)=\frac{1}{4}\).
03

Evaluate the Cosine of the Angle

To evaluate \(\cos(\cos^{-1}\frac{1}{4})\), we substitute \(\theta\) from Step 2 into the cosine function: \[ \cos(\theta) = \cos(\cos^{-1} \frac{1}{4}) \]
04

Simplify the Expression

From Step 2, we found that \(\cos(\theta)=\frac{1}{4}\). So, we can simply substitute the value of \(\cos(\theta)\) back into the expression: \[ \cos(\cos^{-1} \frac{1}{4}) = \frac{1}{4} \] This is the final answer.

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