# What is the general solution to the equation $\sin x + \sqrt{3}\cos x = \sqrt2$

I need to find the general solution to the equation

$$\sin(x) + \sqrt3\cos(x)=\sqrt2$$

So I went ahead and divided by $$2$$, thus getting the form

$$\cos(x-\frac{\pi}{6})=\cos(\frac{\pi}{4})$$

Thus the general solution to this would be $$x = 2n\pi \pm\frac{\pi}{4}+\frac{\pi}{6}$$

Which simplifies out to be,

$$x = 2n\pi +\frac{5\pi}{12}$$ $$x = 2n\pi -\frac{\pi}{12}$$

But the answer doesn't have the 2nd solution as a solution to the given equation. Did I go wrong somewhere?

• Your answer seems to be the correct one. For example $x=-\frac {\pi} {12}$ does satisfy the given equation. Mar 5 '20 at 6:39
• You solution is correct. May be they skip the second one. Mar 5 '20 at 6:43

As Kavi Rama Murthy's comment indicates, you haven't done anything wrong that I can see. You can quite easily very that $$x = 2n\pi - \frac{\pi}{12}$$ is a solution (coming from using $$\cos\left(-\frac{\pi}{4}\right)$$ on the right), as well as the first one you specify of $$x = 2n\pi + \frac{5\pi}{12}$$ (coming from using $$\cos\left(\frac{\pi}{4}\right)$$ on the right). Thus, it seems the answer has an oversight.
• @Techie5879 Unless there's some stated restriction on what $x$ could be, they are both valid options, so it seems the multiple-choice test has a mistake in it. Mar 5 '20 at 6:42