1. The problem asks for the domain and range of all trigonometric functions. 2. The primary trigonometric functions are sine ($\sin x$), cosine ($\cos x$), and tangent ($\tan x$). Additionally, there are cosecant ($\csc x$), secant ($\sec x$), and cotangent ($\cot x$). 3. Domain and range for each: - $\sin x$: - Domain: all real numbers, $(-\infty, \infty)$ - Range: $[-1,1]$ - $\cos x$: - Domain: all real numbers, $(-\infty, \infty)$ - Range: $[-1,1]$ - $\tan x$: - Domain: all real numbers except $x = \frac{\pi}{2} + k\pi$, where $k$ is any integer (points where cosine is zero) - Range: all real numbers, $(-\infty, \infty)$ - $\csc x = \frac{1}{\sin x}$: - Domain: all real numbers except $x = k\pi$, where $k$ is any integer (points where sine is zero) - Range: $(-\infty, -1] \cup [1, \infty)$ - $\sec x = \frac{1}{\cos x}$: - Domain: all real numbers except $x = \frac{\pi}{2} + k\pi$, where $k$ is any integer (points where cosine is zero) - Range: $(-\infty, -1] \cup [1, \infty)$ - $\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$: - Domain: all real numbers except $x = k\pi$, where $k$ is any integer (points where sine is zero) - Range: all real numbers, $(-\infty, \infty)$ These domain restrictions are due to division by zero in reciprocal functions or undefined points in $\tan$ and $\cot$. Final summary: - $\sin x$: domain $(-\infty, \infty)$, range $[-1,1]$ - $\cos x$: domain $(-\infty, \infty)$, range $[-1,1]$ - $\tan x$: domain $x \neq \frac{\pi}{2} + k\pi$, range $(-\infty, \infty)$ - $\csc x$: domain $x \neq k\pi$, range $(-\infty, -1] \cup [1, \infty)$ - $\sec x$: domain $x \neq \frac{\pi}{2} + k\pi$, range $(-\infty, -1] \cup [1, \infty)$ - $\cot x$: domain $x \neq k\pi$, range $(-\infty, \infty)$