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 $2\pi$ \[ \sin(\theta+2\pi)=\sin\theta,\qquad\cos(\theta+2\pi)=\cos\theta \] but the tangent and cotangent functions have a smaller period $\pi$, because (see Equation 18(c) in Trignometric Identities) \[ \tan(\theta+\pi)=\tan\theta,\qquad\cot(\theta+\pi)=\cot\theta. \]

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 $\sin0=0$. 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.

Figure 2: Graph of $y=\sin\theta$ in one period

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

Figure 3: Graph of $y=\sin\theta$. Domain$=(-\infty,\infty)$, range $=[-1,1]$, period$=2\pi$

  •  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 $\cos0=1$ 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 $\cos2\pi=1$ (Figure 4(a)), and it will start all over again (Figure 4(b)).

(a) (b)Domain $=(-\infty,\infty)$, 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 $\cos\theta=\sin\left(\theta+\frac{\pi}{2}\right)$ (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 $\{y|-1\leq y\leq1\}=[-1,1]$.
  • The sine function takes on 0 at $x=0,\pm\pi,\pm2\pi,\pm3\pi,\cdots$.
  • 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 $\tan0=0$. 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 \[\theta\neq(2k+1)\frac{\pi}{2},\quad k\text{ is an integer.}\]
  • As we can see in Figure 6, the range of $y=\tan\theta$ is the entire set of real numbers $\mathbb{R}$ or $(-\infty,\infty)$.
  • 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 $=\{\theta|\ \theta\neq(2k+1)\frac{\pi}{2},\ k\in\mathbb{Z}\}$

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

Example 1
Given the graph of $y=\tan x$, use transformations to graph $y=\cot x$, and then determine its domain and its range.
Solution
We know \[ \tan\left(\frac{\pi}{2}-x\right)=\cot x. \] 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 \[ x\neq0,\pm\pi,\pm2\pi,\pm3\pi,\cdots \] or \[ Dom(\cot x)=\{x|\ x\neq k\pi,k\in\mathbb{Z}\} \] and its range is \[ \{y|\ -\infty<y<\infty\} \] 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 $y=\tan x$.

Example 2
Use the graph of $y=\sin x$ to plot $y=3\sin2x$.
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 $y=\sin2x$ (Figure 8(a)). Then we vertically scale it by a fractor of 3, to get the graph of $y=3\sin2x$ (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