Domain, range and graphs

Now that the six ratios are defined for every angle, we can treat them as func­tions of a real num­ber $x$ (an angle in radi­ans) and ask the usual ques­tions about func­tions: for which $x$ are they defined (the domain), what val­ues can they take (the range), and what do their graphs look like? The graphs make many facts obvi­ous, such as why $\sin x$ never exceeds $1$, and they let us com­pare val­ues like $\sin 1$, $\sin 2$ and $\sin 3$ with­out a cal­cu­la­tor.

Domain and range#

The rea­sons are sim­ple. Since $|y| \le r$ and $|x| \le r$, sine and cosine lie between $-1$ and $1$. Tan­gent and secant have $x$ (that is, $\cos x$) in the denom­i­na­tor, so they are unde­fined where $\cos x = 0$, at the odd mul­ti­ples of $\displaystyle \frac{\pi}{2}$. Cotan­gent and cose­cant have $\sin x$ in the denom­i­na­tor, so they are unde­fined at the mul­ti­ples of $\pi$. Secant and cose­cant are rec­i­p­ro­cals of num­bers between $-1$ and $1$, so they are never strictly between $-1$ and $1$.

Peri­odic func­tions#

A func­tion is peri­odic with period $T$ if its val­ues repeat every $T$: $f(x + T) = f(x)$ for all $x$. Since adding $2\pi$ brings an angle back to the same posi­tion, sine, cosine, secant and cose­cant have period $2\pi$. Tan­gent and cotan­gent repeat sooner, because $\tan(\pi + x) = \tan x$: their period is $\pi$.

The graph of sin x#

Plot $y = \sin x$ from these val­ues:

The curve is a smooth wave: it rises from $0$ to its high­est value $1$ at $\displaystyle \frac{\pi}{2}$, falls through $0$ at $\pi$ to its low­est value $-1$ at $\displaystyle \frac{3\pi}{2}$, and returns to $0$ at $2\pi$. The same wave repeats every $2\pi$ to the left and right. It crosses the x-axis at $x = n\pi$.

Near the ori­gin the graph hugs the line $y = x$. In fact, for $\displaystyle 0 < x < \frac{\pi}{2}$,

$$\sin x < x < \tan x$$

For instance at $\displaystyle x = \frac{\pi}{3} \approx 1.047$: $\sin x \approx 0.866$ and $\tan x \approx 1.732$. For $\displaystyle -\frac{\pi}{2} < x < 0$ the inequal­i­ties reverse: $\tan x < x < \sin x$.

The graph of cos x#

It is the same wave as $\sin x$, shifted $\displaystyle \frac{\pi}{2}$ to the left: it starts at its high­est point $1$ when $x = 0$. It crosses the x-axis at $\displaystyle x = (2n + 1)\frac{\pi}{2}$, and its period is $2\pi$.

The graph of tan x#

$\tan x = 0$ at $x = n\pi$, and it is unde­fined at $\displaystyle x = (2n + 1)\frac{\pi}{2}$. Between $\displaystyle -\frac{\pi}{2}$ and $\displaystyle \frac{\pi}{2}$ the graph rises steadily from very large neg­a­tive val­ues, through $(0, 0)$ and $\displaystyle \left(\frac{\pi}{4}, 1\right)$, to very large pos­i­tive val­ues. The ver­ti­cal lines $\displaystyle x = \pm\frac{\pi}{2}$, which the curve approaches but never meets, are called asymp­totes. The same branch repeats every $\pi$.

The graphs of cot x, sec x and csc x#

• $\cot x$: unde­fined at $x = n\pi$ (ver­ti­cal asymp­totes there), zero at $\displaystyle x = (2n + 1)\frac{\pi}{2}$, and falling on each branch. Period $\pi$.
• $\sec x$: unde­fined where $\cos x = 0$. Where $\cos x$ is near $1$, $\sec x$ is near $1$; as $\cos x$ shrinks to $0$, $\sec x$ grows with­out limit. The graph is a row of U-shaped branches, open­ing upwards above $y = 1$ and down­wards below $y = -1$. Period $2\pi$.
• $\csc x$: the same shape as $\sec x$, but with asymp­totes at $x = n\pi$. Period $2\pi$.

Com­par­ing val­ues with the graphs#

Remem­ber that $1$ radian $\approx 57^\circ$, $2$ radi­ans $\approx 115^\circ$ and $3$ radi­ans $\approx 172^\circ$, while $\displaystyle \frac{\pi}{2} \approx 1.57$ and $\pi \approx 3.14$.

Exam­ple 1#

Which is greater, $\sin 1^\circ$ or $\sin 1$?

$1$ radian is about $57^\circ$. On $\displaystyle \left[0, \frac{\pi}{2}\right]$ the sine graph rises, so the larger angle has the larger sine: $\sin 1 > \sin 1^\circ$.

Exam­ple 2#

Arrange $\sin 1$, $\sin 2$ and $\sin 3$ in order.

All three angles lie between $0$ and $\pi$, where sine is pos­i­tive. On the graph, $\sin x$ is high­est at $\displaystyle \frac{\pi}{2} \approx 1.57$ and falls away on both sides. The angle $2$ is $0.43$ from $\displaystyle \frac{\pi}{2}$, the angle $1$ is $0.57$ from it, and $3$ is $1.43$ from it, almost at $\pi$. So

$$\sin 2 > \sin 1 > \sin 3$$

(The val­ues are about $0.909$, $0.841$ and $0.141$.)

Exam­ple 3#

Arrange $\cos 1$, $\cos 2$ and $\cos 3$ in order.

On $[0, \pi]$ the cosine graph falls steadily from $1$ to $-1$. So $\cos 1 > \cos 2 > \cos 3$.

Exam­ple 4#

Arrange $\tan 1$, $\tan 2$ and $\tan 3$ in order.

$\displaystyle 1 < \frac{\pi}{2}$, so $\tan 1$ is pos­i­tive. The angles $2$ and $3$ lie between $\displaystyle \frac{\pi}{2}$ and $\pi$, where tan­gent is neg­a­tive, and on that branch tan­gent rises, so $\tan 2 < \tan 3 < 0$. Hence

$$\tan 1 > \tan 3 > \tan 2$$

Prac­tice#

1. Which of these are impos­si­ble? (a) $\sin\theta = 1.2$ (b) $\cos\theta = -0.4$ (c) $\tan\theta = 100$ (d) $\sec\theta = 0.5$ (e) $\csc\theta = -3$
2. Write the period of $\sin x$, $\cos x$ and $\tan x$.
3. Find the great­est and least val­ues of $3\sin x$, and of $5 - 2\cos x$.
4. Write the domain of $\sec x$.
5. Which is greater, $\sin 2^\circ$ or $\sin 2$?
6. Arrange $\cos 0.5$, $\cos 1.5$ and $\cos 2.5$ in decreas­ing order.
7. Arrange $\sin 1$, $\sin 1.5$ and $\sin 2.5$ in decreas­ing order.
8. For which val­ues of $x$ in $[0, 2\pi]$ is $\tan x$ not defined?

Answers#