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

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Solution

$We have : \sin x + \cos x = 0 \Rightarrow \sin x = - \cos x \Rightarrow \frac{\sin x}{\cos x} = - 1 \Rightarrow \tan x = - 1$

$Now, x is in the fourth quadrant . In the fourth quadrant, \cos x and \sec x are positive and all the other four T - ratios are negative .$

$∴ s e c x = \sqrt{1 + \tan^{2} x} = \sqrt{1 + {(- 1)}^{2}} = \sqrt{2} \cos x = \frac{1}{s e c x} = \frac{1}{\sqrt{2}} And, \sin x = - \sqrt{1 - \cos^{2} x} = - \sqrt{1 - {(\frac{1}{\sqrt{2}})}^{2}} = \frac{- 1}{\sqrt{2}} ∴ \sin x = \frac{- 1}{\sqrt{2}} and \cos x = \frac{1}{\sqrt{2}}$

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