Sure, let's go through this step-by-step to find the exact value of the given expression:
[tex]\[ \arccos \left[\cos \left(-\frac{9 \pi}{2}\right)\right] \][/tex]
### Step 1: Understand the periodicity of the cosine function
The cosine function is periodic with a period of [tex]\(2\pi\)[/tex]. This means:
[tex]\[ \cos(\theta) = \cos(\theta + 2k\pi) \][/tex]
for any integer [tex]\( k \)[/tex].
### Step 2: Simplify the angle [tex]\(-\frac{9\pi}{2}\)[/tex]
We need to reduce [tex]\(\theta = -\frac{9\pi}{2}\)[/tex] to an equivalent angle within the range [tex]\([0, 2\pi)\)[/tex] or [tex]\([- \pi, \pi )\)[/tex].
To do this, we add or subtract multiples of [tex]\(2\pi\)[/tex]. Let's find the closest equivalent angle:
First, we can see:
[tex]\[ -\frac{9\pi}{2} + 4\pi = -\frac{9\pi}{2} + \frac{8\pi}{2} = -\frac{\pi}{2} \][/tex]
### Step 3: Calculate the cosine value
Next, we evaluate:
[tex]\[ \cos\left(-\frac{\pi}{2}\right) \][/tex]
We know from trigonometric properties:
[tex]\[ \cos\left(-\frac{\pi}{2}\right) = 0 \][/tex]
### Step 4: Calculate the arccosine value
Now, we need:
[tex]\[ \arccos(0) \][/tex]
The arccosine function, [tex]\(\arccos(x)\)[/tex], gives us the angle whose cosine is [tex]\(x\)[/tex], and it ranges from [tex]\(0\)[/tex] to [tex]\(\pi\)[/tex].
Since:
[tex]\[ \cos\left(\frac{\pi}{2}\right) = 0 \][/tex]
We have:
[tex]\[ \arccos(0) = \frac{\pi}{2} \][/tex]
### Conclusion
The exact value of the given expression is:
[tex]\[ \arccos \left[\cos \left(-\frac{9 \pi}{2}\right)\right] = \frac{\pi}{2} \][/tex]
Thus, the exact result is:
[tex]\[ \boxed{\frac{\pi}{2}} \][/tex]
