Question

If $\cos \theta =\frac{12}{13}$ and $\theta$ is acute, what is the value of $\text{cosec}\theta$?

• A. $\frac{12}{5}$
• B. $\frac{5}{13}$
• C. $\frac{13}{5}$
• D. $\frac{11}{5}$

Answer 1

Answer: (C)

$\frac{13}{5}$

First use the identity: ${\sin }^2\theta +{\cos }^2\theta =1$ and substitute the value of $\cos \theta =\frac{12}{13}$

Therefore ${\left(\frac{12}{13}\right)}^2+{\sin }^2\theta =1$

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

Squaring on both the sides, we get

${\sin }^2\theta =\sqrt{\frac{25}{169}}=\frac{5}{13}$

Now we know the value of ${\sin }^2\theta$, to find $\text{cosec}{}^2\theta$ we just take the reciprocal.

Therefore, $\text{cosec}{}^2\theta =\frac{13}{5}$

So, option $C$ is correct.