#### Answer

We can find the domain of $sin ~\theta$:
domain: $-\infty \lt \theta \lt \infty$
We can find the domain of $cos ~\theta$:
domain: $-\infty \lt \theta \lt \infty$
We can find the domain of $tan ~\theta$:
domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$
We can find the domain of $csc ~\theta$:
domain: $\theta \neq \pi~n$, for any integer $n$
We can find the domain of $sec ~\theta$:
domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$
We can find the domain of $cot ~\theta$:
domain: $\theta \neq \pi~n$, for any integer $n$

#### Work Step by Step

We can find the domain of $sin ~\theta$:
$sin~\theta = \frac{x}{r}$
Since $r$ is always positive, $sin ~\theta$ is defined for all real numbers.
domain: $-\infty \lt \theta \lt \infty$
We can find the domain of $cos ~\theta$:
$cos~\theta = \frac{y}{r}$
Since $r$ is always positive, $cos ~\theta$ is defined for all real numbers.
domain: $-\infty \lt \theta \lt \infty$
We can find the domain of $tan ~\theta$:
$tan~\theta = \frac{sin~\theta}{cos~\theta}$
$tan ~\theta$ is undefined when $cos~\theta = 0$
$tan ~\theta$ is undefined when $\theta = \frac{\pi}{2}+\pi~n$, for any integer $n$
domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$
We can find the domain of $csc ~\theta$:
$csc~\theta = \frac{1}{sin~\theta}$
$csc ~\theta$ is undefined when $sin~\theta = 0$
$csc ~\theta$ is undefined when $\theta = \pi~n$, for any integer $n$
domain: $\theta \neq \pi~n$, for any integer $n$
We can find the domain of $sec ~\theta$:
$sec~\theta = \frac{1}{cos~\theta}$
$sec ~\theta$ is undefined when $cos~\theta = 0$
$sec ~\theta$ is undefined when $\theta = \frac{\pi}{2}+\pi~n$, for any integer $n$
domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$
We can find the domain of $cot ~\theta$:
$cot~\theta = \frac{cos~\theta}{sin~\theta}$
$cot ~\theta$ is undefined when $sin~\theta = 0$
$cot ~\theta$ is undefined when $\theta = \pi~n$, for any integer $n$
domain: $\theta \neq \pi~n$, for any integer $n$