Question

Evaluate \[ \cos \frac{7\pi}{12}. \]

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

Hint:

Express angles as sums of standard angles (\(\pi/3, \pi/4, \pi/6\)) and use sum formula for cosine.

Answer 1

The Correct Option is C

Step 1: Write the angle as a difference of two standard angles.
Instead of splitting \(\frac{7\pi}{12} = 105^\circ\) as \(60^\circ+45^\circ\), it is cleaner to write it as \(105^\circ = 135^\circ - 30^\circ\), so
\[ \cos 105^\circ = \cos(135^\circ-30^\circ) = \cos135^\circ\cos30^\circ + \sin135^\circ\sin30^\circ \]

Step 2: Substitute the known values.
\[ \cos135^\circ = -\frac{\sqrt{2}}{2}, \quad \cos30^\circ = \frac{\sqrt{3}}{2}, \quad \sin135^\circ = \frac{\sqrt{2}}{2}, \quad \sin30^\circ = \frac{1}{2} \]

Step 3: Compute the value.
\[ \cos105^\circ = \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \]

Step 4: Write the final value.
\[ \boxed{\frac{\sqrt{2}-\sqrt{6}}{4}} \]