Question

Consider the information given below $\tan x=-\frac{5}{12}$, where $x$ lies in second quadrant.
Statement I The values of $\cot x,\sin x,\cos x$, $\mathrm{cosec}x,\sec x$ are $\frac{-12}{5},\frac{5}{13},\frac{-12}{13},\frac{13}{5},\frac{-13}{12}$, respectively.
Statement II In second quadrant the value of $\sin x$ and $\mathrm{cosec}x$ 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)

In second quadrant, $\frac{\pi }{2}<x<\pi$
Here, $\tan x=-\frac{5}{12}$

$\therefore \cot x=\frac{1}{\tan x}=-\frac{12}{5}$
(since, in second quadrant $\sin x$ and $\mathrm{cosec}x$ are positive)
$\therefore \sin x=\frac{5}{13},\mathrm{cosec}x=\frac{13}{5},\cos x=\frac{-12}{13},\sec x=\frac{-13}{12}$
Alternate Method
$\tan x=-\frac{5}{12}$
Given that, $x$ lies in second quadrant, i.e., $\frac{\pi }{2}<x<\pi$
$\cot x=\frac{1}{\tan x}=-\frac{12}{5}$
Now, ${\sec }^{2}x=1+{\tan }^{2}x=1+{\left(-\frac{5}{12}\right)}^2$
$=1+\frac{25}{144}=\frac{169}{144}$
$\Rightarrow {\sec }^{2}x={\left(\frac{13}{12}\right)}^2$
$\Rightarrow \sec x=\pm \frac{13}{12}$
$\because x$ lies in second quadrant, so we take negative sign.
$\therefore \sec x=-\frac{13}{12}$
$\Rightarrow \cos x=\frac{1}{\sec x}=\frac{-12}{13}$
Now, $\tan x=\frac{\sin x}{\cos x}$
$\Rightarrow -\frac{5}{12}=\frac{\sin x}{-\frac{12}{13}}$
$\Rightarrow \sin x=\frac{5}{13}$
and $\mathrm{cosec}x=\frac{1}{\sin x}$
$=\frac{13}{5}$
Hence, option (b) is correct.