Domains and Ranges

Domains

The domain is which angles you can plug in. For most trig functions, it’s “everything except where the denominator is zero.”

Sine and cosine work for any angle. The point on the unit circle always has an x and y coordinate.

Domain of $\sin \theta$ and $\cos \theta$: all real numbers

Tangent $= \sin / \cos$ breaks when $\cos \theta = 0$.

That happens at $90°, 270°, 450°, \ldots$ In general:

Domain of $\tan \theta$: all reals except $\theta = \frac{\pi}{2} + n\pi$, where $n$ is any integer

Cosecant $= 1 / \sin$ breaks when $\sin \theta = 0$.

That happens at $0°, 180°, 360°, \ldots$

Domain of $\csc \theta$: all reals except $\theta = n\pi$

Secant $= 1 / \cos$, same restrictions as tangent:

Domain of $\sec \theta$: all reals except $\theta = \frac{\pi}{2} + n\pi$

Cotangent $= \cos / \sin$, same restrictions as cosecant:

Domain of $\cot \theta$: all reals except $\theta = n\pi$

Notice the pairs: $\tan$ and $\sec$ share the same excluded points (both have $\cos$ in the denominator). $\csc$ and $\cot$ share the same excluded points (both have $\sin$ in the denominator).

Ranges

The range is what values can come out.

Sine and Cosine

The unit circle has radius 1. The x-coordinate can’t go past 1 or below $-1$. Same for y.

Range of $\sin \theta$ and $\cos \theta$: $[-1, 1]$

Tangent and Cotangent

As $\theta$ approaches $90°$, the terminal side gets steeper and steeper. The slope grows without bound.

Range of $\tan \theta$ and $\cot \theta$: all real numbers, $(-\infty, \infty)$

Cosecant and Secant

Since $\sin \theta$ is between $-1$ and $1$, its reciprocal $\csc \theta = 1 / \sin \theta$ is either $\geq 1$ or $\leq -1$.

It can never land between $-1$ and $1$, because $1$ divided by a number between $-1$ and $1$ always gives something with absolute value greater than $1$.

Range of $\csc \theta$ and $\sec \theta$: $(-\infty, -1] \cup [1, \infty)$

Summary