Question

Find the value of the other five trigonometric functions 
\[\tan x = \frac{3}{4},\] x in quadrant III

Answer 1

 We have:
\[\tan x = \frac{3}{4}\text{ and }x\text{ are in the third quadrant.}\]
\[\text{ In the third quadrant, }\tan x,\text{ and }\cot x \text{ are positive . }\]
\[\text{ And, }\sin x, \cos x , \sec x\text{ and }cosec x \text{ are negative .}\]
\[\therefore \cot x = \frac{1}{\tan x} = \frac{1}{\frac{3}{4}} = \frac{4}{3}\]
\[\sec x = - \sqrt{1 + \tan^2 x} = - \sqrt{1 + \left( \frac{3}{4} \right)^2} = - \frac{5}{4}\]
\[\cos x = \frac{1}{\sec x} = \frac{1}{- \frac{5}{4}} = - \frac{4}{5}\]
\[\sin x = - \sqrt{1 - \cos^2 x} = - \sqrt{1 - \left( \frac{- 4}{5} \right)^2} = \frac{- 3}{5}\]
\[cosec x = \frac{1}{\sin x} = \frac{1}{- \frac{3}{5}} = - \frac{5}{3}\]