Question

If \(\sin \theta = \frac{1}{9}\), then \(\tan \theta\) is equal to

• A. \(\frac{1}{4\sqrt{5}}\)
• B. \(\frac{4\sqrt{5}}{9}\)
• C. \(\frac{1}{8}\)
• D. \(4\sqrt{5}\)

Answer 1

The Correct Option is A

Given:
\[ \sin \theta = \frac{1}{9} \]

Step 1: Find \(\cos \theta\) using identity
\[ \sin^2 \theta + \cos^2 \theta = 1 \] \[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{1}{9}\right)^2 = 1 - \frac{1}{81} = \frac{80}{81} \] \[ \cos \theta = \sqrt{\frac{80}{81}} = \frac{\sqrt{80}}{9} = \frac{4\sqrt{5}}{9} \]

Step 2: Calculate \(\tan \theta\)
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{9}}{\frac{4\sqrt{5}}{9}} = \frac{1}{4\sqrt{5}} \]

Final Answer:
\[ \boxed{ \tan \theta = \frac{1}{4\sqrt{5}} } \]