Question

Find the values of other five trigonometric functions if $cot\,x=\frac{3}{4},$ x lies in third quadrant

Answer 1

$cot\,x=\frac{3}{4}$

$cot\,x\,=\frac{1}{cot\,x}=\frac{1}{(\frac{3}{4})}=\frac{4}{3}$

$1+tan^2=sec^2\,x$

$⇒1+(\frac{4}{3})^2=sec^2\ x$

$⇒1+\frac{16}{9}=sec^2\ x$

$⇒=\frac{25}{9}=sec^2x$

$⇒sec\,x=±\frac{5}{3}$

Since x lies in the 3rd quadrant, the value of sec x will be negative.

$∴sec\,x=-\frac{5}{3}$

$cos\,x=\frac{1}{sec\,\,x}=\frac{1}{(-\frac{5}{3})}=-\frac{3}{5}$

$tan\,x=\frac{sin\,x}{cos \,x}$

$⇒\frac{4}{3}=\frac{sin\,x}{(\frac{-3}{5})}$

$⇒sin\,x=(\frac{4}{3})×(\frac{-3}{5})=-\frac{4}{5}$

$cosec\,x=\frac{1}{sin\,x}=-\frac{5}{4}$