3.3.4 Periodicity and Graphs of Trigonometric Functions

Periodicity of Trigonometric Functions

All trigonometric functions are defined in terms of the cosine and sine functions. The cosine and sine functions are characterized as the $x$ and $y$ coordinates of the intersection point $P$ of the terminal side of the angle $\theta$ and the unit circle (Figure 1).

Figure 1

Because one revolution corresponds to $2\pi$ radians or 360 degrees, the same point $P$ is obtained for $\theta +2\pi k$ where $k$ is an integer. Thus the values of the trigonometric functions repeat when $\theta$ increases by $2\pi$. The sine and cosine functions are periodic with period $2\pi$
$$\sin (\theta +2\pi )=\sin \theta ,\cos (\theta +2\pi )=\cos \theta$$ but the tangent and cotangent functions have a smaller period $\pi$, because (see Equation 18(c) in Trignometric Identities)
$$\tan (\theta +\pi )=\tan \theta ,\cot (\theta +\pi )=\cot \theta .$$

We may use the periodicity of trigonometric functions to sketch their graphs. So we can just sketch the graphs of sine and cosine functions for $\theta$ between $0$ and $2\pi$ (or for $\theta$ between $-\pi$ and $\pi$ or in general for $\theta$ between $a$ and $a+2\pi$ for any real number $a$) and then obtain the graph for all real values of $\theta$ by infinitely many repetitions of this cycle, to the right and to the left.

Graph of y = sin 𝜽

To sketch the graph of $\sin \theta$, we just need to look at Figure 2 and follow how the $y$ coordinate of the point $P$ varies as $\theta$ increases from $0$ to $2\pi$ and $P$ completes one counterclockwise revolution. We know $\sin 0=0$. As $\theta$ goes from $0$ to $\pi /2$, $\sin \theta$ increases until it reaches $+1$ at $\theta =\pi /2$. As $\theta$ goes from $\pi /2$ to $\pi$, $\sin \theta$ decreases until reaches $0$ at $\theta =\pi$. When $\theta$ is between $\pi$ to $3\pi /2$, $\sin \theta$ is negative and keep decreasing until it reaches $-1$ at $\theta =3\pi /2$. Finally as $\theta$ goes from $3\pi /2$ to $2\pi$, $\sin \theta$ increases from $-1$ to $0$ and will be ready to start all over again.

The complete graph $y=\sin \theta$ consists of infinitely many repetitions of one cycle just shown (Figure 3).

•  Note that $y=\sin \theta$ is an odd function (that is, $\sin (-\theta )=-\sin \theta ),$and thus its graph is symmetric about the origin.

Graph of y = cos 𝜽

We can graph $y=\cos \theta$ essentially the same way. The main difference is $\cos 0=1$ and $\cos \theta$ decreases until it becomes $\cos \frac{\pi }{2}=0$ and finally it reaches $-1$ at $\theta =\pi$. At this point, $\cos \theta$ starts to increase until it becomes $\cos \frac{3\pi }{2}=0$ and then $\cos 2\pi =1$ (Figure 4(a)), and it will start all over again (Figure 4(b)).

(a)	(b)Domain $=(-\mathrm{\infty },\mathrm{\infty })$, range$=[-1,1]$, period$=2\pi$

Figure 4: Graph of $y=\cos \theta$ (a) in one cycle (b) in several cycles

Another way of drawing the graph of $y=\cos \theta$ is by shifting the graph of $y=\sin \theta$ to the left $\pi /2$ units (Figure 5) because $\cos \theta =\sin (\theta +\frac{\pi }{2})$ (see Equation 17(a) in Trigonometric Identites). As we can see in Figure 5:

• The sine and cosine functions are defined for all values of $\theta$ and their range is $\{y|-1\le y\le 1\}=[-1,1]$.
• The sine function takes on 0 at $x=0,\pm \pi ,\pm 2\pi ,\pm 3\pi ,\cdots$.
• The cosine function takes on 0 at $x=\pm \frac{\pi }{2},\pm \frac{3\pi }{2},\pm \frac{5\pi }{2},\pm \frac{7\pi }{2},\cdots$

Figure 5: Graph of $y=\cos \theta$ is obtained by shifting the graph of $y=\sin \theta$ to the left $\pi /2$ units.

Note that $y=\cos \theta$ is an even function (that is, $\cos (-\theta )=\cos \theta )$, and thus its graph is symmetric about the $y$-axis.

Graph of y = tan 𝜽

Now let’s sketch the graph of $y=\tan \theta$. We discussed that the period of this function is $\pi$, so we can plot it in one period and find the complete graph by repeating what we obtain in one interval. For example, we can take the interval
$$-\frac{\pi }{2}<\theta <\frac{\pi }{2}.$$ Note that $\tan \theta =\sin \theta /\cos \theta$ is not defined for $\theta =\pm \pi /2$ because because $\cos (\pm \pi /2)=0$ and division by zero is not defined. We know $\tan 0=0$. When $-\frac{\pi }{2}<\theta <0$ (that is when $\theta$ is in the fourth quadrant), $\tan \theta <0$ and when $0<\theta <\frac{\pi }{2}$, $\tan \theta >0$. As $\theta$ increases toward $\pi /2$, $\tan \theta$ is positive and becomes indefinitely great, while as $\theta$ decreases toward $-\frac{\pi }{2},$ $\tan \theta$ is negative and becomes indefinitely great. You can also see this in Figure 6(a).

•  The domain of $y=\tan \theta$ is the set of all $\theta$-values such that
  $$\theta \ne (2k+1)\frac{\pi }{2},k\text{ is an integer.}$$
• As we can see in Figure 6, the range of $y=\tan \theta$ is the entire set of real numbers $\mathbb{R}$ or $(-\mathrm{\infty },\mathrm{\infty })$.
• To graph trigonometric functions in the coordinate plane, as usual, we often denote the independent variable by $x$. So we often talk about the graphs of $y=\sin x,y=\cos x$, $y=\tan x$, etc.

(a)	(b) Domain $=\{\theta |\theta \ne (2k+1)\frac{\pi }{2},k\in \mathbb{Z}\}$

Figure 6: (a) Graph of $y=\tan \theta$ in one period $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$. (b) The complete graph of $y=\tan \theta$.

Examples

Solution

We know
$$\tan (\frac{\pi }{2}-x)=\cot x.$$ So we start with the graph of $y=\tan x$ (Figure 7(a)). Then we can plot the graph of $y=f(x)$, where
$$f(x)=\tan (-x),$$ by reflecting the graph of $y=\tan x$ in the $y$-axis (Figure 7(b)). Now we can get the graph of $y=f(x+\pi /2)=\tan (-x+\pi /2)=\cot x$ by shifting the graph of $f$ to the left $\pi /2$ units (Figure 7(c)). As we can see $\cot x$ is defined for all $x$-values such that
$$x\ne 0,\pm \pi ,\pm 2\pi ,\pm 3\pi ,\cdots$$ or
$$Dom(\cot x)=\{x|x\ne k\pi ,k\in \mathbb{Z}\}$$ and its range is
$$\{y|-\mathrm{\infty }<y<\mathrm{\infty }\}$$ or the entire set of real numbers $\mathbb{R}$.

(a) Graph of $y=\tan x$	(b) Graph of $y=f(x)=\tan (-x)$	(c) Graph of $y=f(x+\pi /2)=\cot x$

Figure 7: Sketching graph of $y=\cot x$ by transforming the graph of $y=\tan x$.

Solution

Recall that to obtain the graph of $y=f(cx)$ for $c>1$, we horizontally compress the graph of $y=f(x)$ by a fractor of $1/c$. Thus here we horizontally compress the graph of $y=\sin x$ by a factor of 1/2 to obtain the graph of $y=\sin 2x$ (Figure 8(a)). Then we vertically scale it by a fractor of 3, to get the graph of $y=3\sin 2x$ (Figure 8(b)). We see that when $x$ is a multiple of $\pi /2$, $y=0$ and when $x$ is an odd multiple of $\pi /4$, $y=3$ or $y=-3$. For all other values of $x$, $-3<y<3$.

(a)	(b)

Figure 8