Find this function range $f(x)=\sqrt{2+\cos{x}+\sqrt{3}\sin{x}}+2\sqrt{2+\cos{x}-\sqrt{3}\sin{x}}$

Question

let $x\in R$, find the following function range $$f(x)=\sqrt{2+\cos{x}+\sqrt{3}\sin{x}}+2\sqrt{2+\cos{x}-\sqrt{3}\sin{x}}$$

since use wolfarmapha see:wolf

I Found no parsing value, but this is a test questions, I think should have good results, the possible problem is what happened to the software settings, lead to can't get the answer we need

Answer 1

Hint:

Using $\cos2y=2\cos^2y-1$

$$2+\cos x+\sqrt3\sin x=2\left(1+\cos\left(x-\dfrac\pi3\right)\right)=4\cos^2\left(\dfrac x2-\dfrac\pi6\right)$$

$$2+\cos x-\sqrt3\sin x=2\left(1+\cos\left(x+\dfrac\pi3\right)\right)=4\cos^2\left(\dfrac x2+\dfrac\pi6\right)$$

Now $\sqrt{a^2}=\begin{cases} a &\mbox{if } a\ge0 \\-a & \mbox{if } a<0\end{cases}$

Answer 2

Using

$$ a \cos x \pm b \sin x = \sqrt{a^2+b^2}\cos(x \pm \arctan(b/a)) $$

we have

$$f(x)=\sqrt{2+\cos{x}+\sqrt{3}\sin{x}}+2\sqrt{2+\cos{x}-\sqrt{3}\sin{x}} = \sqrt{2}\left(\sqrt{1+\cos(x+\pi/3))}+2\sqrt{1+\cos(x-\pi/3)}\right)$$

etc.