Find the values of other five trigonometric functions if $\cot x=\frac{3}{4}, x$ lies in third quadrant.
$\cot x=\frac{3}{4}$
$\tan x=\frac{1}{\cot x}=\frac{1}{\left(\frac{3}{4}\right)}=\frac{4}{3}$
$1+\tan ^{2} x=\sec ^{2} x$
$\Rightarrow 1+\left(\frac{4}{3}\right)^{2}=\sec ^{2} x$
$\Rightarrow 1+\frac{16}{9}=\sec ^{2} x$
$\Rightarrow \frac{25}{9}=\sec ^{2} x$
$\Rightarrow \sec x=\pm \frac{5}{3}$
since $x$ lies in the $3^{\text {rd }}$ quadrant, the value of sec $x$ will be negative.
$\therefore \sec x=-\frac{5}{3}$
$\cos x=\frac{1}{\sec x}=\frac{1}{\left(-\frac{5}{3}\right)}=-\frac{3}{5}$
$\tan x=\frac{\sin x}{\cos x}$
$\Rightarrow \frac{4}{3}=\frac{\sin x}{\left(\frac{-3}{5}\right)}$
$\Rightarrow \sin x=\left(\frac{4}{3}\right) \times\left(\frac{-3}{5}\right)=-\frac{4}{5}$
$\cos ec\, x=\frac{1}{\sin x}=-\frac{5}{4}$
If $\cot x=-\frac{5}{12}, x$ lies in second quadrant, find the values of other five trigonometric functions.
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