Find the exact value of: 𝑐𝑜𝑠(11π/12)

To find the exact value of \( \cos\left( \frac{11\pi}{12} \right) \), we can use the **sum or difference identity** for cosine.

### Step 1: Express \( \frac{11\pi}{12} \) as a sum or difference
We can rewrite \( \frac{11\pi}{12} \) as:
\[
\frac{11\pi}{12} = \pi - \frac{\pi}{12}
\]
This gives us a difference:
\[
\cos\left( \frac{11\pi}{12} \right) = \cos\left( \pi - \frac{\pi}{12} \right)
\]

### Step 2: Apply the cosine difference identity
The identity for \( \cos(\pi - x) \) is:
\[
\cos(\pi - x) = -\cos(x)
\]
So:
\[
\cos\left( \frac{11\pi}{12} \right) = -\cos\left( \frac{\pi}{12} \right)
\]

### Step 3: Find \( \cos\left( \frac{\pi}{12} \right) \)
We can now use the half-angle identity to find \( \cos\left( \frac{\pi}{12} \right) \). Since \( \frac{\pi}{12} = \frac{\pi}{6}/2 \), we apply the half-angle formula:
\[
\cos\left( \frac{\theta}{2} \right) = \sqrt{\frac{1 + \cos(\theta)}{2}}
\]
Let \( \theta = \frac{\pi}{6} \), then:
\[
\cos\left( \frac{\pi}{12} \right) = \sqrt{\frac{1 + \cos\left( \frac{\pi}{6} \right)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}
\]

### Step 4: Simplify the expression
First, simplify the expression inside the square root:
\[
\frac{1 + \frac{\sqrt{3}}{2}}{2} = \frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2} = \frac{2 + \sqrt{3}}{4}
\]
Thus:
\[
\cos\left( \frac{\pi}{12} \right) = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2}
\]

### Step 5: Final answer
Since \( \cos\left( \frac{11\pi}{12} \right) = -\cos\left( \frac{\pi}{12} \right) \), we get:
\[
\cos\left( \frac{11\pi}{12} \right) = -\frac{\sqrt{2 + \sqrt{3}}}{2}
\]

Thus, the exact value is:
\[
\cos\left( \frac{11\pi}{12} \right) = -\frac{\sqrt{2 + \sqrt{3}}}{2}
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the value of \( \cos( \frac{7\pi}{12} ) \)?
2. How does the sum identity for cosine differ from the difference identity?
3. Can you use the unit circle to verify \( \cos(\pi - x) = -\cos(x) \)?
4. What is the sine value for \( \frac{11\pi}{12} \)?
5. How do you derive the half-angle identity for cosine?

**Tip**: When working with trigonometric functions, breaking down angles into familiar values often simplifies the problem significantly.

Learn how to find the exact value of cos(11π/12) using trigonometric identities. This step-by-step solution applies the cosine difference identity and the half-angle identity, breaking down the process for high school students in Grades 10-12.

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