`cos^(-1)(cos"(5pi)/4)=`

Options

A. `-pi/4`
B. `pi/4`
C. `(3pi)/4`
D. `(5pi)/4`

Answer

To solve the expression \( \cos^{-1}(\cos(5\pi/4)) \), we will follow these steps:

### Step 1: Identify the Principal Value Range
The principal value of the inverse cosine function, \( \cos^{-1}(x) \), is defined in the range \( [0, \pi] \).

### Step 2: Analyze the Angle
The angle \( 5\pi/4 \) is greater than \( \pi \). To find an equivalent angle within the principal value range, we can express \( 5\pi/4 \) in terms of \( \pi \):
\[ 5\pi/4 = \pi + \pi/4 \]

### Step 3: Use the Cosine Addition Formula
Using the property of cosine:
\[ \cos(\pi + \theta) = -\cos(\theta) \]
we can rewrite:
\[ \cos(5\pi/4) = \cos(\pi + \pi/4) = -\cos(\pi/4) \]

### Step 4: Find the Cosine Value
Now, we need to find \( \cos(\pi/4) \):
\[ \cos(\pi/4) = \frac{\sqrt{2}}{2} \]
Thus,
\[ \cos(5\pi/4) = -\frac{\sqrt{2}}{2} \]

### Step 5: Substitute Back into the Inverse Function
Now we substitute this back into the inverse cosine function:
\[ \cos^{-1}(\cos(5\pi/4)) = \cos^{-1}(-\frac{\sqrt{2}}{2}) \]

### Step 6: Determine the Angle for the Inverse Cosine
The angle whose cosine is \( -\frac{\sqrt{2}}{2} \) within the range \( [0, \pi] \) is:
\[ \theta = \frac{3\pi}{4} \]

### Final Answer
Thus, we have:
\[ \cos^{-1}(\cos(5\pi/4)) = \frac{3\pi}{4} \]

### Summary
The final answer is:
\[ \frac{3\pi}{4} \]