Question

Find the value of the other five trigonometric functions 

\[\cot x = \frac{12}{5},\] x in quadrant III

Answer 1

 We have:
\[\cot x = \frac{12}{5} \text{ and }x \text{ are in the third quadrant. }\]
\[\text{ In the third quadrant,} \tan x \text{ and }\cot x\text{ are positive }\]
\[\text{ And, }\sin x, \cos x , \sec x\text{ and cosec x are negative. }\]
\[\therefore \tan x = \frac{1}{\cot x} = \frac{1}{\frac{12}{5}} = \frac{5}{12}\]
\[cosec x = - \sqrt{1 +\cot^2 x} = - \sqrt{1 + \left( \frac{12}{5} \right)^2} = - \frac{13}{5}\]
\[\sin x = \frac{1}{cosecx} = \frac{1}{- \frac{13}{5}} = - \frac{5}{13}\]
\[\cos x = - \sqrt{1 - \sin^2 x} = - \sqrt{1 - \left( \frac{- 5}{13} \right)^2} = \frac{- 12}{13}\]
\[\text{ And, }\sec x = \frac{1}{\cos x} = \frac{1}{\frac{- 12}{13}} = \frac{- 13}{12}\]