If \( \theta \) lies in the fourth quadrant and \( 3 \cot \theta + 4 = 0 \), then what is the value of \( \sin 2\theta + \cos 2\theta \) ?
- A. -\frac{31}{25}
- B. -\frac{17}{25} ✓
- C. 0
- D. 1
Correct Answer: B. -\frac{17}{25}
Explanation
Given 3 \cot \theta + 4 = 0 \implies \cot \theta = -4/3. In the 4th quadrant, \cos \theta = 4/5 and \sin \theta = -3/5. Using double angle formulas: \sin 2\theta = 2(-3/5)(4/5) = -24/25, and \cos 2\theta = (4/5)^2 - (-3/5)^2 = 7/25. Their sum is -17/25.