Question

If \( \sin \alpha = \frac{12}{13} \), where \( \frac{\pi}{2}<\alpha<\frac{3\pi}{2} \), then the value of \( \tan \alpha \) is equal to

• A. \( \frac{5}{12} \)
• B. \( \frac{13}{5} \)
• C. \( -\frac{12}{5} \)
• D. \( -\frac{13}{5} \)
• E. \( -\frac{1}{12} \)

Hint:

Always check quadrant to assign correct sign for trigonometric functions.

Answer 1

The Correct Option is C

Concept: \[ \sin^2\alpha + \cos^2\alpha = 1 \]

Step 1: Find \( \cos \alpha \).
\[ \cos\alpha = \pm \sqrt{1 - \sin^2\alpha} = \sqrt{1 - \left(\frac{12}{13}\right)^2} = \frac{5}{13} \] Since \( \frac{\pi}{2}<\alpha<\frac{3\pi}{2} \Rightarrow \cos\alpha<0 \), \[ \cos\alpha = -\frac{5}{13} \]

Step 2: Compute \( \tan \alpha \).
\[ \tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{12/13}{-5/13} = -\frac{12}{5} \]