Question

If \(\sin \theta = \frac{1}{3}\), then \(\sec \theta\) is equal to:

• A. \(\frac{2\sqrt{2}}{3}\)
• B. \(\frac{3}{2\sqrt{2}}\)
• C. \(3\)
• D. \(\frac{1}{\sqrt{3}}\)

Answer 1

The Correct Option is B

Step 1: Given information
We are given that \( \sin \theta = \frac{1}{3} \).

Step 2: Use the identity to find \( \cos \theta \)
We use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substitute \( \sin \theta = \frac{1}{3} \):
\[ \left(\frac{1}{3}\right)^2 + \cos^2 \theta = 1 \Rightarrow \frac{1}{9} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{1}{9} = \frac{8}{9} \Rightarrow \cos \theta = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \]

Step 3: Use identity to find \( \sec \theta \)
We know that \( \sec \theta = \frac{1}{\cos \theta} \)
\[ \sec \theta = \frac{1}{\frac{2\sqrt{2}}{3}} = \frac{3}{2\sqrt{2}} \]

Final Answer:
\[ \sec \theta = \frac{3}{2\sqrt{2}} \]