Without the aid of a calculator, calculate the exact value of cos11𝜋12

We are asked to find the exact value of \( \cos\left(\frac{11\pi}{12}\right) \). This angle lies in the second quadrant, where cosine is negative. To simplify the calculation, we can express the angle as a sum or difference of standard angles.

### Step-by-step solution:

1. **Express the angle as a sum of known angles**:
   \[
   \frac{11\pi}{12} = \pi - \frac{\pi}{12}
   \]
   This rewrites the angle as:
   \[
   \cos\left(\frac{11\pi}{12}\right) = \cos\left(\pi - \frac{\pi}{12}\right)
   \]
   Using the identity \( \cos(\pi - \theta) = -\cos(\theta) \), we get:
   \[
   \cos\left(\frac{11\pi}{12}\right) = -\cos\left(\frac{\pi}{12}\right)
   \]

2. **Now find \( \cos\left(\frac{\pi}{12}\right) \)**:
   We can use the half-angle identity for cosine:
   \[
   \cos\left(\frac{\pi}{12}\right) = \cos\left(15^\circ\right)
   \]
   We use the identity for cosine of half-angles:
   \[
   \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}
   \]
   Let \( \theta = \frac{\pi}{6} = 30^\circ \), so:
   \[
   \cos\left(\frac{\pi}{12}\right) = \sqrt{\frac{1 + \cos\left(\frac{\pi}{6}\right)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}}
   \]
   Simplifying:
   \[
   \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2 + \sqrt{3}}}{2}
   \]

3. **Find the exact value**:
   Using the result from step 2, we substitute back:
   \[
   \cos\left(\frac{11\pi}{12}\right) = -\frac{\sqrt{2 + \sqrt{3}}}{2}
   \]

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

Would you like further details or have any questions?

Here are some related questions to extend your understanding:
1. How do you derive the half-angle identity for cosine?
2. What is the exact value of \( \sin\left(\frac{11\pi}{12}\right) \)?
3. How would you calculate \( \cos\left(\frac{7\pi}{12}\right) \)?
4. What are the general properties of trigonometric functions in different quadrants?
5. How can you use the sum or difference identities to solve other trigonometric equations?

**Tip:** Trigonometric identities, like the half-angle and sum/difference formulas, are invaluable for finding exact values of non-standard angles!

Without the aid of a calculator, calculate the exact value of cos(11π/12).

Learn how to find the exact value of cos(11π/12) without a calculator by applying trigonometric identities and half-angle formulas. This step-by-step solution covers key concepts in trigonometry, including angle reduction and cosine identities, suitable for Grades 10-12.

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