Question

Considering only the principal values of the inverse trigonometric functions, evaluate: \[ \cot^{-1}(\cot(-11)) +10\sin\left(2\cos^{-1}\left(\frac1{\sqrt2}\right)\right) +10\sin\left(2\tan^{-1}(2)\right) \]

• A. \(3\pi+7\)
• B. \(7\)
• C. \(4\pi+7\)
• D. \(3\pi-5\)

Hint:

Principal value range: \[ \cot^{-1}x\in(0,\pi) \] Useful identity: \[ \sin(2\theta)=\frac{2\tan\theta}{1+\tan^2\theta} \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We must evaluate each term of the expression by strictly following the principal value ranges:
- \( \cot^{-1}x \in (0, \pi) \)
- \( \cos^{-1}x \in [0, \pi] \)
- \( \tan^{-1}x \in (-\pi/2, \pi/2) \)
For the first term, we need to find an angle in \( (0, \pi) \) whose cotangent is equal to \( \cot(-11) \).
Step 3: Detailed Explanation:
1. Evaluating \( \cot^{-1}(\cot(-11)) \):
Let \( \cot^{-1}(\cot(-11)) = \theta \). Then \( \cot \theta = \cot(-11) \) and \( \theta \in (0, \pi) \).
We know \( \cot(x) = \cot(n\pi + x) \). We need \( -11 + n\pi \in (0, \pi) \).
Approximate \( \pi \approx 3.14 \).
For \( n = 4 \): \( 4(3.14) - 11 = 12.56 - 11 = 1.56 \). Since \( 1.56 \in (0, 3.14) \), our value is \( 4\pi - 11 \).
2. Evaluating \( 10 \sin(2 \cos^{-1}(1/\sqrt{2})) \):
\( \cos^{-1}(1/\sqrt{2}) = \pi/4 \).
The term is \( 10 \sin(2 \cdot \pi/4) = 10 \sin(\pi/2) = 10 \cdot 1 = 10 \).
3. Evaluating \( 10 \sin(2 \tan^{-1}(2)) \):
Let \( \alpha = \tan^{-1}(2) \), then \( \tan \alpha = 2 \).
We need \( 10 \sin(2\alpha) \). Using the identity \( \sin(2\alpha) = \frac{2\tan\alpha}{1+\tan^2\alpha} \):
\( \sin(2\alpha) = \frac{2(2)}{1+2^2} = \frac{4}{5} \).
The term is \( 10 \cdot (4/5) = 8 \).
4. Total sum:
Sum \( = (4\pi - 11) + 10 + 8 = 4\pi + 7 \).
Step 4: Final Answer:
The total value is \( 4\pi + 7 \).