Question

If \(\sin \theta = \frac{12}{13}\),then \(\text{tan}\ θ =\)

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

Answer 1

The Correct Option is D

Correct answer: \(\frac{12}{5}\) 

Explanation:
Given: \[ \sin \theta = \frac{12}{13} \] We can use the identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting the given value of \( \sin \theta \): \[ \left( \frac{12}{13} \right)^2 + \cos^2 \theta = 1 \] \[ \frac{144}{169} + \cos^2 \theta = 1 \] \[ \cos^2 \theta = 1 - \frac{144}{169} = \frac{169}{169} - \frac{144}{169} = \frac{25}{169} \] \[ \cos \theta = \frac{5}{13} \] Now, use the identity for tangent: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} \]

Hence, the value of \( \tan \theta \) is \(\frac{12}{5}\).