Question

If $\sin \theta =\frac{5}{13}$ and $\theta$ is acute, what is the value of $\sec \theta$?

• A. $\frac{2}{13}$
• B. $\frac{13}{12}$
• C. $\frac{12}{13}$
• D. $\frac{1}{13}$

Answer 1

Answer: (B)

$\frac{13}{12}$

First use the identity: ${\sin }^2\theta +{\cos }^2\theta =1$ and substitute the value of $\sin \theta =\frac{5}{13}$

Therefore, $(\frac{5}{13}{)}^2+{\cos }^2\theta =1$

${\cos }^2\theta =1-\frac{25}{169}=\frac{144}{169}$

Squaring on both the sides, we get

${\cos }^2\theta =\sqrt{\frac{144}{169}}=\frac{12}{13}$

Now we know the value of ${\cos }^2\theta$, to find ${\sec }^2\theta$

Therefore, ${\sec }^2\theta =\frac{13}{12}$

So, option $B$ is correct.