Range of f(x) = $\frac{\sqrt3\,\sin x}{2 + \cos x}$ Can you give any idea about the range of the following function?
$$f(x) = \frac{\sqrt{3}\,\sin x}{2 + \cos x}$$

Consider $f^2(x) = \dfrac{3 - 3cos^2x}{cos^2x + 4cosx + 4}$. Now let $t = cosx$, then look at $f(t) = \dfrac{3 - 3t^2}{t^2 + 4t + 4}$. We find $f_{max}$. $f'(t) = \dfrac{-6(2t + 1)}{t + 2} = 0$ when $t = \dfrac{-1}{2}$. So $f(-1) = 0$, $f(\frac{-1}{2}) = 1$, and $f(1) = 0$. So $f_{max}^2 = 1$. This means that: $f^2(x) \le 1$, and so: $-1 \le f(x) \le 1$. So the range of $f$ is: $[-1, 1]$
