Question

Find the values of other five trigonometric functions if  \(sec\, x=\frac{13}{5}\), x lies in fourth quadrant.

Answer 1

\(sec\,x=\frac{13}{5}\)

\(cot\,x\,=\frac{1}{sec\,x}=\frac{1}{(\frac{13}{5})}=\frac{5}{13}\)

\(⇒sin^2x+cos^2x=1\)

\(⇒sin^2x=1-cos^2x\)

\(⇒sin^2x=1-(\frac{5}{13})^2\)

\(⇒sin^2x=1-\frac{25}{169}=\frac{144}{169}\)

\(⇒sin\,x=±\frac{12}{13}\)

Since x lies in the 4^th quadrant, the value of sin x will be negative.

\(sin\,x=-\frac{12}{13}\)

\(cosec\,x=\frac{1}{sin\,\,x}=\frac{1}{(-\frac{12}{13})}=-\frac{13}{12}\)

\(tax\,\,x=\frac{sin\,x}{cos\,\,x}=\frac{(\frac{-12}{13})}{(\frac{5}{13})}=-\frac{12}{5}\)

\(cot\,x=\frac{1}{tan\ x}=\frac{1}{(-\frac{12}{5})}=-\frac{5}{12}\)