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

We are given \(\sin \theta = \frac{1}{3}\). We need to find \(\sec \theta\).

Recall the following trigonometric identities: \(\sec \theta = \frac{1}{\cos \theta}\)

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

From \(\sin \theta = \frac{1}{3}\), we can find \(\cos \theta\) using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\[ \left(\frac{1}{3}\right)^2 + \cos^2 \theta = 1 \implies \frac{1}{9} + \cos^2 \theta = 1 \]

\[ \cos^2 \theta = 1 - \frac{1}{9} = \frac{8}{9} \]

Thus:

\[ \cos \theta = \frac{\sqrt{8}}{3} \]

Now, we can find \(\sec \theta\):

\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{8}}{3}} = \frac{3}{\sqrt{8}} = \frac{3}{2\sqrt{2}} \]

Thus, the correct answer is:

\(b)\ \frac{3}{2\sqrt{2}}\)