Trigonometry – Problem 17: find the range of

Range and Domain Relationship

Given $f(x) = \sin\left(\omega x + \frac{\pi}{6}\right)$ with domain $[m, n]$ ($m < n$) and range $[0, 1]$, find the range of $n - m$.

For $x \in [m, n]$, we have $\omega x + \frac{\pi}{6} \in \left[\omega m + \frac{\pi}{6}, \omega n + \frac{\pi}{6}\right]$.

For range $[0, 1]$, the argument must cover from $0$ to $\frac{\pi}{2}$ (at minimum) but less than a full period.

Thus:

$$\frac{\pi}{2} \leq \left(\omega n + \frac{\pi}{6}\right) - \left(\omega m + \frac{\pi}{6}\right) < \pi$$

$$\frac{\pi}{2} \leq \omega(n - m) < \pi$$

For $\omega = 1$: $n - m \in \left[\frac{\pi}{2}, \pi\right)$

$\left[\frac{\pi}{2}, \pi\right)$