Question

If $\text{cosec}\theta =\frac{13}{5}$ and $\theta \notin 1^{\text{st}}\text{quadrant}$. Then the value of $\sec \theta$ is

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

Answer 1

The correct option is D $-\frac{13}{12}$

$\text{cosec}\theta =\frac{13}{5}\Rightarrow \sin \theta =\frac{5}{13}$
$\Rightarrow \cos \theta =\pm \sqrt{1-{\sin }^2\theta }=\pm \sqrt{1-{\left(\frac{5}{13}\right)}^2}$
$\Rightarrow \cos \theta =\pm \frac{12}{13}$
$\because \text{cosec}\theta$ is $+ive$ but $\theta \notin 1^{\text{st}}\text{quadrant}$
$\therefore \theta \in 2^{\text{nd}}$ quadrant
Hence, $\cos \theta =-\frac{12}{13}$
$\Rightarrow \sec \theta =-\frac{13}{12}$