Domains, ranges and graphs of trigonometric functions

Each trigono­met­ric ratio is a func­tion of the angle: put in an angle, get out a num­ber. In this les­son you will find the domain and range of all six trigono­met­ric func­tions, see what their graphs look like, and learn their peri­ods (how often each graph repeats). The graphs then answer ques­tions such as "which is greater, $\sin 1$ or $\sin 1°$?" with­out any cal­cu­la­tion.

The range of $\sin\theta$#

Take $P(x, y)$ on the arm of $\theta$ with $OP = r$, so $x^2 + y^2 = r^2$. The point $P$ lies on a cir­cle of radius $r$, so its $y$-coor­di­nate can be any­thing from $-r$ (at the bot­tom, $(0, -r)$) to $r$ (at the top, $(0, r)$):

$$\displaystyle -r \leq y \leq r \implies -1 \leq \frac{y}{r} \leq 1 \implies -1 \leq \sin\theta \leq 1, \quad \forall \theta \in \mathbb{R}$$

Every angle has a sine, so

$$\text{Domain of } \sin\theta = \mathbb{R}, \qquad \text{Range} = [-1, 1]$$

The same argu­ment with $x$ gives the same domain and range for $\cos\theta$.

The domain and range of $\tan\theta$#

$\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$ is defined only where $\cos\theta \neq 0$, that is, $\displaystyle \theta \neq (2n + 1)\frac{\pi}{2}$:

$$\displaystyle \text{Domain of } \tan\theta = \mathbb{R} - \left\{(2n + 1)\frac{\pi}{2} : n \in \mathbb{Z}\right\}$$

Since $\displaystyle \tan\theta = \frac{y}{x}$, it can take any real value: with $x = 1$, $y = 10$ we get $\tan\theta = 10$; with $x = 1$, $y = 100$ we get $100$; and so on, pos­i­tive or neg­a­tive. So the range of $\tan\theta$ is $\mathbb{R}$.

Table of domains and ranges#

The domains of $\tan$ and $\sec$ leave out the zeros of $\cos x$; the domains of $\cot$ and $\csc$ leave out the zeros of $\sin x$. The range of $\sec x$ comes from turn­ing $[-1, 1]$ upside down: the rec­i­p­ro­cal of a num­ber between $-1$ and $1$ (other than $0$) is at least $1$ in size. So $\sec x$ and $\csc x$ never take a value strictly between $-1$ and $1$; their graphs lie out­side the band between the lines $y = 1$ and $y = -1$.

The graphs and their peri­ods#

$\sin x$. A smooth wave through the ori­gin. It rises to $1$ at $\displaystyle \frac{\pi}{2}$, falls back to $0$ at $\pi$, down to $-1$ at $\displaystyle \frac{3\pi}{2}$ and back to $0$ at $2\pi$; to the left it reaches $-1$ at $\displaystyle -\frac{\pi}{2}$ and $1$ at $\displaystyle -\frac{3\pi}{2}$. The pat­tern repeats every $2\pi$: the period of $\sin x$ is $2\pi$.

$\cos x$. The same wave shifted: max­i­mum $1$ at $0$, zero at $\displaystyle \pm\frac{\pi}{2}$, min­i­mum $-1$ at $\pm\pi$. Period $2\pi$.

$\tan x$. Sep­a­rate branches, each ris­ing from very large neg­a­tive val­ues to very large pos­i­tive val­ues. One branch passes through $0$, another through $\pi$. Between branches there are ver­ti­cal lines, called asymp­totes, at $\displaystyle -\frac{\pi}{2}$, $\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{3\pi}{2}$, ..., which the curve approaches but never meets. Period $\pi$.

$\cot x$. Decreas­ing branches, cross­ing the $x$-axis at $\displaystyle \frac{\pi}{2}$ and $\displaystyle \frac{3\pi}{2}$, with asymp­totes at $0$, $\pi$, $2\pi$. Period $\pi$.

$\sec x$. U-shaped branches: one open­ing upwards with low­est point $1$ at $x = 0$, between the asymp­totes $\displaystyle -\frac{\pi}{2}$ and $\displaystyle \frac{\pi}{2}$; the next open­ing down­wards with high­est point $-1$ at $x = \pi$, between $\displaystyle \frac{\pi}{2}$ and $\displaystyle \frac{3\pi}{2}$. Period $2\pi$.

$\csc x$. Sim­i­lar U-shapes: low­est point $1$ at $\displaystyle \frac{\pi}{2}$ between asymp­totes $0$ and $\pi$, high­est point $-1$ at $\displaystyle \frac{3\pi}{2}$ between $\pi$ and $2\pi$. Period $2\pi$.

Using the graphs to com­pare val­ues#

Exam­ple 1#

If $x = \sin 1$ and $y = \sin 1°$, which is greater?

$\sin 1$ means the sine of $1$ radian, and $1^c \approx 57°$. So $x = \sin 57°$ (about) and $y = \sin 1°$. On the graph of $\sin$, the curve rises all the way from $0°$ to $90°$, so a larger angle in this range has a larger sine. Since $57° > 1°$,

$$\sin 1° < \sin 1, \quad \text{that is,} \quad x > y$$

Exam­ple 2#

If $x = \tan 1$, $y = \tan 2$, $z = \tan 3$, arrange them in order.

Change to degrees: $1^c \approx 57°$, $2^c \approx 115°$, $3^c \approx 172°$.

• $\tan 57°$ is pos­i­tive (first quad­rant), and larger than $\tan 45° = 1$.
• $\tan 115°$ and $\tan 172°$ are neg­a­tive (sec­ond quad­rant).
• On the branch of $\tan$ from $90°$ to $180°$ the curve rises from very large neg­a­tive val­ues up to $0$. So $\tan 115°$ is a large neg­a­tive num­ber, while $\tan 172°$ is a small neg­a­tive num­ber close to $0$.

There­fore $\tan 1 > \tan 3 > \tan 2$, that is, $x > z > y$.

Exam­ple 3#

Which of these are pos­si­ble? (a) $\displaystyle \sin\theta = \frac{3}{2}$ (b) $\displaystyle \sec\theta = \frac{1}{2}$ (c) $\tan\theta = 100$.

(a) Not pos­si­ble: $\sin\theta$ lies in $[-1, 1]$, and $\displaystyle \frac{3}{2} > 1$.

(b) Not pos­si­ble: $\sec\theta$ is never strictly between $-1$ and $1$.

(c) Pos­si­ble: the range of $\tan\theta$ is all of $\mathbb{R}$.

Exam­ple 4#

Find the range of $3\sin x + 2$.

From $-1 \leq \sin x \leq 1$, mul­ti­ply by $3$: $-3 \leq 3\sin x \leq 3$. Add $2$: $-1 \leq 3\sin x + 2 \leq 5$. The range is $[-1, 5]$.

Prac­tice#

1. Is $\sin\theta = -0.8$ pos­si­ble?
2. Is $\displaystyle \cos\theta = \frac{5}{4}$ pos­si­ble?
3. Is $\sec\theta = 0.6$ pos­si­ble?
4. Is $\csc\theta = -3$ pos­si­ble?
5. Write the domain of $\cot x$.
6. Find the range of $2\cos x - 1$.
7. Find the range of $5 - 3\sin x$.
8. Which is greater, $\cos 1$ or $\cos 1°$?
9. Arrange $\sin 1$, $\sin 2$, $\sin 3$ in decreas­ing order.
10. Write the peri­ods of $\tan x$ and of $\sec x$.
11. For which $x$ in $[0, 2\pi]$ is $\tan x$ not defined?

Answers#