Question

Find the values of other five trigonometric functions if  \(tan\, x=-\frac{5}{12}\), x lies in second quadrant.

Answer 1

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

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

\(1+tan^2x=sec^2x\)

\(⇒1+(-\frac{5}{12})^2=sec^2x\)

\(⇒1+\frac{25}{144}=sec^2x\)

\(⇒\frac{169}{144}=sec^2x\)

\(∴sec\,x=±\frac{13}{12}\)

Since x lies in the 2^nd quadrant, the value of sec x will be negative.

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

\(cos\,x=\frac{1}{sec\,x}=\frac{1}{(-\frac{13}{12})}=-\frac{12}{13}\)

\(tan\,x=\frac{sin\,x}{cos\,\,x}\)

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

\(⇒sin\,x=(-\frac{5}{12})×(-\frac{12}{13})={\frac{5}{13}}\)

\(cosec\,x=\frac{1}{sin\,x}=\frac{1}{(\frac{5}{13})}=\frac{13}{5}\)