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: Understanding the given value:
We are given that \( \sin \theta = \frac{1}{3} \), and we need to find the value of \( \sec \theta \).
Recall that \( \sec \theta \) is the reciprocal of \( \cos \theta \), i.e.,
\[ \sec \theta = \frac{1}{\cos \theta} \] To find \( \sec \theta \), we first need to find \( \cos \theta \).

Step 2: Using the Pythagorean identity:
We can use the Pythagorean identity for sine and cosine, which is:
\[ \sin^2 \theta + \cos^2 \theta = 1 \] Substitute \( \sin \theta = \frac{1}{3} \) into the identity:
\[ \left( \frac{1}{3} \right)^2 + \cos^2 \theta = 1 \] Simplifying the expression:
\[ \frac{1}{9} + \cos^2 \theta = 1 \] Now subtract \( \frac{1}{9} \) from both sides:
\[ \cos^2 \theta = 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \] Taking the square root of both sides:
\[ \cos \theta = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3} \] (Note: We assume \( \theta \) is in the first quadrant, so \( \cos \theta \) is positive.)

Step 3: Finding \( \sec \theta \):
Now that we know \( \cos \theta = \frac{2\sqrt{2}}{3} \), we can find \( \sec \theta \) as the reciprocal of \( \cos \theta \):
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{2\sqrt{2}}{3}} = \frac{3}{2\sqrt{2}} \]

Step 4: Conclusion:
Thus, \( \sec \theta = \frac{3}{2\sqrt{2}} \).