Trigonometric Graphs & Properties

Find the domain of $y = \sqrt{\frac{1}{2} - \cos x}$.

Given $f(x) = \sin\left(\frac{x}{3} + \varphi\right)$ where $0 \leq \varphi < 2\pi$ is an even function, find $\varphi$.

Translate $y = \sin x + \cos x$ left by $m$ units ($0 < m < \pi$). If the resulting graph is symmetric about the y-axis, find $m$.

Find the minimum positive period of $f(x) = \sin\frac{x}{3}\cos\frac{x}{3}$.

Find the minimum positive period of $f(x) = \frac{2\tan x}{1-\tan^2 x}$.

Find the range of $f(x) = \sin^2 x - \cos^2 x$ for $x \in \left[\frac{\pi}{12}, \frac{2\pi}{3}\right]$.

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 $n - m$.

Given $f(x) = \sin\left(\omega x + \frac{\pi}{6}\right)$ ($\omega > 0$). If $f\left(\frac{\pi}{6}\right) = 0$ and $f(x)$ has exactly one zero in $\left[\frac{\pi}{6}, \frac{5\pi}{24}\right]$, find the minimum value of $\omega$.

Given $f(x) = \sin\left(\omega x + \frac{\pi}{3}\right)$ ($\omega > 0$). If $f(x)$ is monotonically increasing on $\left[-\frac{2\pi}{3}, \frac{\pi}{6}\right]$, find the range of $\omega$.

Given $f(x) = 2\cos^2 x(\sin x + \cos x)$ ($\omega > 0$) has its graph symmetric about $x = \frac{\pi}{12}$, and $f(x)$ has no minimum value on $\left[0, \frac{\pi}{3}\right]$. Find $\omega$.