Question

$\frac{\pi}{2} \le \theta \le \frac{3\pi}{4}$ then $\cos^{-1}(\frac{5}{13} \sin \theta + \frac{12}{13} \cos \theta) =$

• A. $\theta - \tan^{-1}(\frac{4}{3})$
• B. $\theta + \tan^{-1}(\frac{5}{12})$
• C. $\theta + \tan^{-1}(\frac{4}{5})$
• D. $\theta - \tan^{-1}(\frac{5}{12})$

Hint:

Always identify the coefficients as $\sin$ and $\cos$ of a common angle to simplify inverse trigonometric expressions.

Answer 1

The Correct Option is D

Step 1: Concept
Rewrite the expression inside $\cos^{-1}$ as a single cosine function using the identity $\cos(A-B) = \cos A \cos B + \sin A \sin B$.

Step 2: Meaning
Let $\cos \alpha = 12/13$ and $\sin \alpha = 5/13$. Then $\tan \alpha = 5/12 \implies \alpha = \tan^{-1}(5/12)$.

Step 3: Analysis
The expression becomes $\cos^{-1}(\sin \alpha \sin \theta + \cos \alpha \cos \theta) = \cos^{-1}(\cos(\theta - \alpha))$.

Step 4: Conclusion
Under the given range of $\theta$, $\cos^{-1}(\cos(\theta - \alpha)) = \theta - \alpha$. Thus, the result is $\theta - \tan^{-1}(5/12)$. Final Answer: (D)