Question

Solve the following equation:
$\sin x+\cos x=\frac{1+\sqrt{3}}{2}$

Answer 1

$\sin x+\cos x=\frac{1+\sqrt{3}}{2}$

Divide by $\sqrt{2}$ on both side, we get

$\sin (x+\frac{\pi }{4})=\frac{1+\sqrt{3}}{2\sqrt{3}}$

Let $\frac{1+\sqrt{3}}{2\sqrt{3}}=a,$

So, $(x+\frac{\pi }{4})=n\pi +{(-1)}^n{\sin }^{-1}a,x=n\pi +{(-1)}^n{\sin }^{-1}a-\frac{\pi }{4}.$