Question

Consider the information given below $\sec x=\frac{13}{5}$ and $x$ lies in fourth quadrant.
Statement I The values of $\cos x,\sin x,\mathrm{cosec}x$, $\tan x,\cot x$ 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., $\sin x=-\frac{12}{13}$
$\tan x=\frac{\sin x}{\cos x}=\frac{-\left(\frac{12}{13}\right)}{\left(\frac{5}{13}\right)}=-\frac{12}{5}$
$\Rightarrow \mathrm{cosec}x=\frac{1}{\sin x}=\frac{-13}{12},\cot x=\frac{1}{\tan x}=-\frac{5}{12}$