Question

Evaluate \[ \cos \frac{\pi}{12}. \]

• A. \(\frac{\sqrt{2}-\sqrt{3}}{2}\)
• B. \(\frac{\sqrt{2}+\sqrt{3}}{2}\)
• C. \(\frac{\sqrt{2}-\sqrt{6}}{4}\)
• D. \(\frac{\sqrt{2}+\sqrt{6}}{4}\)

Hint:

Use sum or difference formulas for cosine to evaluate angles that are not standard multiples of \(\pi/6\) or \(\pi/4\).

Answer 1

The Correct Option is D

Step 1: Express angle in sum form.
\[ \frac{\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6} \]

Step 2: Use cosine difference formula.
\[ \cos(A-B) = \cos A \cos B + \sin A \sin B \]

Step 3: Substitute values.
\[ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin \frac{\pi}{6} = \frac{1}{2} \]

Step 4: Compute using formula.
\[ \cos \frac{\pi}{12} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \]

Step 5: Final conclusion.
\[ \boxed{\frac{\sqrt{2} + \sqrt{6}}{4}} \]