Question

Find the values of other five trigonometric functions if  \(cos\,x=-\frac{1}{2},\) x lies in third quadrant.

Answer 1

\(cos\,x=-\frac{1}{2}\)

\(∴sec\,x=\frac{1}{cos\,x}=\frac{1}{(-\frac{1}{2})}=-2\)

\(sin^2x+cos^2\,x=1\)

\(⇒sin^2+cos^2\,x=1\)

\(⇒sin^2x=1-(-\frac{1}{2})^2\)

\(⇒sin^2x=1-\frac{1}{4}=\frac{3}{4}\)

\(sin^2x=±\frac{±√3}{2}\)

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

\(∴sin\,x=-\frac{√3}{2}\)

\(cosecx=\frac{1}{sin\,x}=\frac{1}{-\frac{√3}{2}}=-\frac{2}{√3}\)

\(tan\,x=\frac{sin\,x}{cos \,x}=\frac{-\frac{√3}{2}}{-\frac{1}{2}}=√3\)

\(cot\,x=\frac{1}{tan\,x}=\frac{1}{√3}\)