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.

$∴sin\,x=-\frac{√3}{2}$

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

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

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