The exact value of `"cos"(3pi)/(4)` is

Options

A. `-1`
B. `-(sqrt(3))/(2)`
C. `-(sqrt(2))/(2)`
D. `-(1)/(2)`.

Explanation

To find the exact value of \( \cos\left(\frac{3\pi}{4}\right) \), we can follow these steps:

### Step 1: Rewrite the angle
We can express \( \frac{3\pi}{4} \) in a different form:
\[ \frac{3\pi}{4} = \pi - \frac{\pi}{4} \]

### Step 2: Use the cosine subtraction identity
Using the identity \( \cos(\pi - \theta) = -\cos(\theta) \), we can rewrite the cosine:
\[ \cos\left(\frac{3\pi}{4}\right) = \cos\left(\pi - \frac{\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) \]

### Step 3: Find \( \cos\left(\frac{\pi}{4}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 4: Substitute back into the equation
Now substitute \( \cos\left(\frac{\pi}{4}\right) \) back into the equation:
\[ \cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

### Step 5: Rationalize the denominator
To express this in a more standard form, we can rationalize the denominator:
\[ -\frac{1}{\sqrt{2}} = -\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = -\frac{\sqrt{2}}{2} \]

### Final Answer
Thus, the exact value of \( \cos\left(\frac{3\pi}{4}\right) \) is:
\[ -\frac{\sqrt{2}}{2} \]

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