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

Answer: (B)

$\frac{3}{2\sqrt{2}}$

Answer 2

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:

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