Question

Find the value of the other five trigonometric functions 

\[\cos x = - \frac{1}{2},\] x in quadrant II

Answer 1

We have: 
\[\cos x = - \frac{1}{2} \text{ and x are in the second quadrant.}\]
\[\text{ In the second quadrant, }\sin x,\text{ and }cosec x\text{ are positive . }\]
\[\text{ And, }\tan x, \cot x , \cos x \text{ and }\sec x \text{ are negative .}\]
\[\therefore \sin x = \sqrt{1 - \cos^2 x} = \sqrt{1 - \left( \frac{- 1}{2} \right)^2} = \frac{\sqrt{3}}{2}\]
\[\tan x = \frac{\sin x}{\cos x} = \frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}} = \sqrt{3}\]
\[cot x=\frac{1}{\tan x}=\frac{1}{- \sqrt{3}}=\frac{- 1}{\sqrt{3}}\]
\[ \sec x = \frac{1}{\cos x} = \frac{1}{\frac{- 1}{2}} = - 2\]
\[cosec x = \frac{1}{\sin x} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\]