Question

If \[ \sin \theta = \frac{-12}{13}, \quad \cos \phi = \frac{-4}{5}, \quad \text{and} \quad \theta, \phi \text{ lie in the third quadrant, then } \tan(\theta - \phi) = \]

• A. \( \frac{-33}{56} \)
• B. \( \frac{-56}{33} \)
• C. \( \frac{56}{33} \)
• D. \( \frac{33}{56} \)

Hint:

To calculate \( \tan(\theta - \phi) \), use the formula for the tangent of a difference and find the individual tangents using trigonometric identities.

Answer 1

The Correct Option is D

Step 1: Use the formula for \( \tan(\theta - \phi) \).
The formula for \( \tan(\theta - \phi) \) is: \[ \tan(\theta - \phi) = \frac{\tan \theta - \tan \phi}{1 + \tan \theta \tan \phi} \]
Step 2: Finding \( \tan \theta \) and \( \tan \phi \).
From \( \sin \theta = \frac{-12}{13} \), we use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find: \[ \cos \theta = \frac{-5}{13}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-12}{-5} = \frac{12}{5} \] Similarly, from \( \cos \phi = \frac{-4}{5} \), we find: \[ \sin \phi = \frac{-3}{5}, \quad \tan \phi = \frac{\sin \phi}{\cos \phi} = \frac{-3}{-4} = \frac{3}{4} \]
Step 3: Applying the formula for \( \tan(\theta - \phi) \).
Substituting the values of \( \tan \theta \) and \( \tan \phi \) into the formula: \[ \tan(\theta - \phi) = \frac{\frac{12}{5} - \frac{3}{4}}{1 + \frac{12}{5} \cdot \frac{3}{4}} \] Simplifying the expression, we get: \[ \tan(\theta - \phi) = \frac{33}{56} \]
Step 4: Conclusion.
Thus, \( \tan(\theta - \phi) = \frac{33}{56} \), which makes option (D) the correct answer.