Question

If sin x + cos x = 0 and x lies in the fourth quadrant, find sin x and cos x.

 

Answer 1

We have: 
\[\sin x + \cos x = 0\]
\[ \Rightarrow \sin x = - \cos x\]
\[ \Rightarrow \frac{\sin x}{\cos x} = - 1\]
\[ \Rightarrow \tan x = - 1\]
Now, x is in thefourth quadrant . 
In thefourth quadrant, cos x and sec x are positive and all the other four T-ratios are negative.
\[\therefore \sec x = \sqrt{1 + \tan^2 x} = \sqrt{1 + \left( - 1 \right)^2} = \sqrt{2}\]
\[\cos x = \frac{1}{secx} = \frac{1}{\sqrt{2}}\]
\[\text{ And, }\sin x = - \sqrt{1 - \cos^2 x} = - \sqrt{1 - \left( \frac{1}{\sqrt{2}} \right)^2} = \frac{- 1}{\sqrt{2}}\]
\[ \therefore \sin x = \frac{- 1}{\sqrt{2}} \text{ and }\cos x = \frac{1}{\sqrt{2}}\]