For this definition we assume that $0 < \theta < \frac{\pi}{2}$ or $0^\circ < \theta < 90^\circ$.
$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
$\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}}$
$\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
$\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$
For this definition $\theta$ is any angle.
$\sin \theta = \frac{y}{1} = y$
$\cos \theta = \frac{x}{1} = x$
$\tan \theta = \frac{y}{x}$
$\csc \theta = \frac{1}{y}$
$\sec \theta = \frac{1}{x}$
$\cot \theta = \frac{x}{y}$
The domain is all the values of $\theta$ that can be plugged into the function.
$\sin \theta$, $\theta$ can be any angle
$\cos \theta$, $\theta$ can be any angle
$\tan \theta$, $\theta \neq \left(n + \frac{1}{2}\right)\pi$, $n = 0, \pm1, \pm2, \dots$
$\csc \theta$, $\theta \neq n\pi$, $n = 0, \pm1, \pm2, \dots$
$\sec \theta$, $\theta \neq \left(n + \frac{1}{2}\right)\pi$, $n = 0, \pm1, \pm2, \dots$
$\cot \theta$, $\theta \neq n\pi$, $n = 0, \pm1, \pm2, \dots$
The period of a function is the number, $T$, such that $f(\theta + T) = f(\theta)$. So, if $\omega$ is a fixed number and $\theta$ is any angle we have the following periods.
$\sin(\omega \theta) \to T = \frac{2\pi}{\omega}$
$\cos(\omega \theta) \to T = \frac{2\pi}{\omega}$
$\tan(\omega \theta) \to T = \frac{\pi}{\omega}$
$\csc(\omega \theta) \to T = \frac{2\pi}{\omega}$
$\sec(\omega \theta) \to T = \frac{2\pi}{\omega}$
$\cot(\omega \theta) \to T = \frac{\pi}{\omega}$
The range is all possible values to get out of the function.
$-1 \leq \sin \theta \leq 1$
$-1 \leq \cos \theta \leq 1$
$-\infty < \tan \theta < \infty$
$\csc \theta \geq 1$ and $\csc \theta \leq -1$
$\sec \theta \geq 1$ and $\sec \theta \leq -1$
$-\infty < \cot \theta < \infty$