Question

Solve the following equations
$\sqrt{3}$ $\sin x+\cos x=1$.

Answer 1

$\sqrt{3}\sin \left(x\right)+\cos x=1$

$\Rightarrow \sqrt{3}\sin x=1-\cos x$

$\Rightarrow$ Square both the sides :-

$3{\sin }^{2}x=1+{\cos }^{2}x-2\cos x$

$3(1-{\cos }^{2}x)=1+{\cos }^{2}x-2\cos x$

$3-3{\cos }^{2}x=1+{\cos }^{2}x-2\cos x$

$\Rightarrow 4{\cos }^{2}x-2\cos x-2=0$

$\Rightarrow 2{\cos }^{2}x-\cos x-1=0$

$\Rightarrow 2{\cos }^{2}x-2\cos x+\cos x-1=0$

$\Rightarrow (2\cos x+1)(\cos x-1)=0$

$\cos x=\frac{-1}{2}\Rightarrow x=\frac{2\pi }{3}+2\pi n,\frac{5\pi }{3}+2\pi n$

$\cos x=1,x=2\pi n$

Solution: $x=\frac{2\pi }{3}+2\pi n,x=\frac{5\pi }{3}+2\pi n,x=2\pi n$.