Solve √3cos x − sin x =1

Question

Solve \(\sqrt{3}\) cosx − sinx =1

Answer 1

We have: \(\sqrt{3 }\)cos x − sin x = 1 

Dividing both sides by 2, we get 

cos x. \(\frac{\sqrt{3}}{2}\) − sin x . \(\frac{1}{2}\) = \(\frac{1}{2}\) 

⇒ cos x . cos \(\frac{\pi}{6}\) − sin x . sin \(\frac{\pi}{6}\) = \(\frac{1}{2}\) 

⇒ cos (x + \(\frac{\pi}{6}\) ) = cos ( \(\frac{\pi}{3}\) ) 

⇒ x + \(\frac{\pi}{6}\) = \(2n\pi\) ± \(\frac{\pi}{3}\)

⇒ x = \(2nx\) ± \(\frac{\pi}{3}\) − \(\frac{\pi}{6}\) 

⇒ x = \(2n\pi\) + \(\frac{\pi}{3}\) − \(\frac{\pi}{6}\) or x = \(2nx\) − \(\frac{\pi}{3}\) −   \(\frac{\pi}{6}\)  

⇒ x = \(2n\pi\) + \(\frac{\pi}{6}\) or x = \(2n\pi\) − \(\frac{\pi}{2}\)