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Trigonometry – I — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – I2 Marks
Question
Find the other trigonometric functions if : $\cot x=\frac{3}{4}$, $x$ lies in the third quadrant.
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Answer

$ \text { We have } \cot x=\frac{3}{4}$
$\therefore \operatorname{cosec}^2 x=1+\cot ^2 x$
$=1+\left(\frac{3}{4}\right)^2$
$=1+\frac{9}{16}$
$=\frac{25}{16}$
$\therefore \operatorname{cosec} x= \pm \frac{5}{4}$
But $x$ lies in the third quadrant $\therefore \operatorname{cosec} \mathrm{x}$ is negative
$ \therefore \operatorname{cosec} x=-\frac{5}{4}$
$\therefore \sin \mathrm{x}=\frac{1}{\operatorname{cosec} x}=\frac{1}{\left(-\frac{5}{4}\right)}=-\frac{4}{5}$
Now, $\cot x=\frac{\cos x}{\sin x}=\frac{3}{4}$
$\therefore \cos x=\frac{3}{4} \sin x=\frac{3}{4}\left(-\frac{4}{5}\right)=-\frac{3}{5}$
$\sec x=\frac{1}{\cos x}=\frac{1}{\left(-\frac{3}{5}\right)}=-\frac{5}{3} $
$\tan x=\frac{1}{\cot x}=\frac{1}{\frac{3}{4}}=\frac{4}{3}$

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