Find the values of other five trigonometric functions if \(cot\,x=\frac{3}{4},\) x lies in third quadrant
\(cot\,x=\frac{3}{4}\)
\(cot\,x\,=\frac{1}{cot\,x}=\frac{1}{(\frac{3}{4})}=\frac{4}{3}\)
\(1+tan^2=sec^2\,x\)
\(⇒1+(\frac{4}{3})^2=sec^2\ x\)
\(⇒1+\frac{16}{9}=sec^2\ x\)
\(⇒=\frac{25}{9}=sec^2x\)
\(⇒sec\,x=±\frac{5}{3}\)
Since x lies in the 3rd quadrant, the value of sec x will be negative.
\(∴sec\,x=-\frac{5}{3}\)
\(cos\,x=\frac{1}{sec\,\,x}=\frac{1}{(-\frac{5}{3})}=-\frac{3}{5}\)
\(tan\,x=\frac{sin\,x}{cos \,x}\)
\(⇒\frac{4}{3}=\frac{sin\,x}{(\frac{-3}{5})}\)
\(⇒sin\,x=(\frac{4}{3})×(\frac{-3}{5})=-\frac{4}{5}\)
\(cosec\,x=\frac{1}{sin\,x}=-\frac{5}{4}\)