Question

\(\cos\frac{\pi}{12} + \cos\frac{17\pi}{12} + \cos\frac{11\pi}{12}\) is equal to

• A. 1
• B. -1
• C. 0
• D. \(\frac{1-\sqrt{3}}{2\sqrt{2}}\)

Hint:

Convert angles to standard values (like \(75^\circ\)) for easy evaluation.

Answer 1

The Correct Option is D

Concept: Use cosine identities: \[ \cos(\pi+\theta) = -\cos\theta,\quad \cos(\pi-\theta)= -\cos\theta \]

Step 1: Simplify terms.
\[ \cos\frac{17\pi}{12} = \cos\left(\pi+\frac{5\pi}{12}\right) = -\cos\frac{5\pi}{12} \] \[ \cos\frac{11\pi}{12} = \cos\left(\pi-\frac{\pi}{12}\right) = -\cos\frac{\pi}{12} \]

Step 2: Substitute.
\[ \cos\frac{\pi}{12} - \cos\frac{5\pi}{12} - \cos\frac{\pi}{12} \] \[ = -\cos\frac{5\pi}{12} \]

Step 3: Evaluate.
\[ \cos\frac{5\pi}{12} = \cos 75^\circ = \frac{\sqrt{6}-\sqrt{2}}{4} \] \[ \Rightarrow -\cos\frac{5\pi}{12} = -\frac{\sqrt{6}-\sqrt{2}}{4} = \frac{\sqrt{2}-\sqrt{6}}{4} \] \[ = \frac{1-\sqrt{3}}{2\sqrt{2}} \]