Question

The value of \( \cos\left(\frac{\pi}{6}-\cos^{-1}\left(-\frac{\sqrt3}{2}\right)\right) \) is equal to:

• A. \( -\frac{\sqrt1}{2} \)
• B. \( -\frac{1}{\sqrt2} \)
• C. \( -\frac{1}{2} \)
• D. \( \frac{1}{2} \)

Hint:

Always remember the principal value range of inverse cosine: \[ \cos^{-1}x \in [0,\pi] \] When cosine is negative, the angle usually lies in the second quadrant. Important standard values: \[ \cos\frac{\pi}{6}=\frac{\sqrt3}{2} \] \[ \cos\frac{5\pi}{6}=-\frac{\sqrt3}{2} \] Also remember that cosine is an even function: \[ \cos(-\theta)=\cos\theta \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves the interpretation of inverse trigonometric functions with negative arguments.
A key property is that \(\cos^{-1}(-x) = \pi - \cos^{-1}(x)\), where the output is always an angle in the range \([0, \pi]\).
Since the argument \(-\sqrt{3}/2\) is negative, the resulting angle will be in the second quadrant (between \(\pi/2\) and \(\pi\)).
After finding the angle, we subtract it from \(\pi/6\) and evaluate the cosine of the resulting angle using standard identities.
Step 2: Key Formula or Approach:
1. Evaluate \(\theta = \cos^{-1}(-\frac{\sqrt{3}}{2})\).
2. Use the identity \(\cos^{-1}(-x) = \pi - \cos^{-1}(x)\).
3. Use the even property of cosine: \(\cos(-\theta) = \cos(\theta)\).
4. Evaluate the resulting cosine value for the second quadrant.
Step 3: Detailed Explanation:
Let's evaluate the expression step-by-step:
Part 1: Finding \(\cos^{-1}(-\frac{\sqrt{3}}{2})\).
Let \(\alpha = \cos^{-1}(-\frac{\sqrt{3}}{2})\).
By the property of inverse cosine:
\[ \alpha = \pi - \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \]
We know that \(\cos(\pi/6) = \sqrt{3}/2\), so \(\cos^{-1}(\sqrt{3}/2) = \pi/6\).
\[ \alpha = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]
Part 2: Substituting into the main expression.
The expression is \(\cos(\frac{\pi}{6} - \alpha)\):
\[ \text{Expression} = \cos \left( \frac{\pi}{6} - \frac{5\pi}{6} \right) \]
\[ = \cos \left( \frac{-4\pi}{6} \right) \]
\[ = \cos \left( -\frac{2\pi}{3} \right) \]
Part 3: Simplifying the cosine of a negative angle.
Since cosine is an even function (\(\cos(-x) = \cos(x)\)):
\[ \cos \left( -\frac{2\pi}{3} \right) = \cos \left( \frac{2\pi}{3} \right) \]
Part 4: Final Evaluation.
The angle \(2\pi/3\) is in the second quadrant. We can write it as \(\pi - \pi/3\):
\[ \cos \left( \frac{2\pi}{3} \right) = \cos \left( \pi - \frac{\pi}{3} \right) \]
Using the reduction formula \(\cos(\pi - \theta) = -\cos \theta\):
\[ = -\cos \left( \frac{\pi}{3} \right) = -\frac{1}{2} \]
Step 4: Final Answer:
The final value is \(-1/2\).
This matches Option (C).