A problem on range of a trigonometric function: what is the range of $\frac{\sqrt{3}\sin x}{2+\cos x}$? What is the range of the function 
$$\frac{\sqrt{3}\sin x}{2+\cos x}$$ 

Note that  $y=f(x)=\dfrac{\sqrt 3 \sin x}{2+\cos x}$.
Note that  $y^2=\dfrac{3 \sin ^2 x}{(2+\cos ^2 x)^2}=\dfrac{3 (1- \cos ^2 x)}{(2+\cos ^2 x)^2}$
Put $\cos x =u$, we have 
\begin{align}
y^2(2+u)^2 =3(1-u^2) \\ \implies (3+y^2)u^2+4y^2u+4y^2-3=0
\end{align}
For this to have a solution, we need discriminant of the quadratic to be non-negative. Hence, we get
$$(4y)^2-4(3+y^2)(4y^2-3) \ge 0 \\ \implies y^2  \le 1\\ \implies-1 \le y \le 1$$ 
