Question

Consider the information given below $\sec x=\frac{13}{5}$ and $x$ lies in fourth quadrant.
Statement I The values of $\cos x, \sin x, \operatorname{cosec} x$, $\tan x, \cot x$ are $\frac{5}{13}, \frac{-12}{13}, \frac{-13}{12}, \frac{-12}{5}, \frac{-5}{12}$, respectively.
Statement II In fourth quadrant the values of $\cos x$ and $\sec x$ are positive.
Choose the correct option.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (B)

$\sec x=\frac{13}{5}$
Given that $x$ lies in fourth quadrant.
i.e., $ \frac{3 \pi}{2} < x < 2 \pi $
$\Rightarrow \cos x =\frac{5}{13} $
$\because \sin ^2 x+\cos ^2 x =1 $
$\Rightarrow \sin ^2 x =1-\cos ^2 x$
$ =1-\left(\frac{5}{13}\right)^2$
$ =1-\frac{25}{169}$
$ =\frac{169-25}{169}=\frac{144}{169}=\left(\frac{12}{13}\right)^2 $
$\Rightarrow \sin x =\pm \frac{12}{13}$
Since, $x$ lies in fourth quadrant, so we take only negative sign.
i.e., $ \sin x=-\frac{12}{13} $
$ \tan x=\frac{\sin x}{\cos x}=\frac{-\left(\frac{12}{13}\right)}{\left(\frac{5}{13}\right)}=-\frac{12}{5} $
$ \Rightarrow \operatorname{cosec} x=\frac{1}{\sin x}=\frac{-13}{12}, \cot x=\frac{1}{\tan x}=-\frac{5}{12}$