Find the values of other five trigonometric functions if $\sec x=\frac{13}{5}, x$ lies in fourth quadrant.
Find the values of other five trigonometric functions if $\sec x=\frac{13}{5}, x$ lies in fourth quadrant.
$\sec x=\frac{13}{5}$
$\cos x=\frac{1}{\sec x}=\frac{1}{\left(\frac{13}{5}\right)}=\frac{5}{13}$
$\sin ^{2} x+\cos ^{2}=1$
$\Rightarrow \sin ^{2} x=1-\cos ^{2} x$
$\Rightarrow \sin ^{2} x=1-\left(\frac{5}{13}\right)^{2}$
$\Rightarrow \sin ^{2} x=1-\frac{25}{169}=\frac{144}{169}$
$\Rightarrow \sin x=\pm \frac{12}{13}$
since $x$ lies in the $4^{\text {th }}$ quadrant, the value of $\sin x$ will be negative.
$\therefore \sin x=-\frac{12}{13}$
$\cos ec \,x=\frac{1}{\sin x}=\frac{1}{\left(-\frac{12}{13}\right)}=-\frac{13}{12}$
$\tan x=\frac{\sin x}{\cos x}=\frac{\left(\frac{-12}{13}\right)}{\left(\frac{5}{13}\right)}=-\frac{12}{5}$
$\cot x=\frac{1}{\tan x}=\frac{1}{\left(-\frac{12}{5}\right)}=-\frac{5}{12}$
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