8.3 Graphs of trig functions  (Page 13/3)

Graphs of trigonometric functions

This section describes the graphs of trigonometric functions.

Graph of $sin\theta$

Graph of $sin\theta$

Complete the following table, using your calculator to calculate the values. Then plot the values with $sin\theta$ on the $y$ -axis and $\theta$ on the $x$ -axis. Round answers to 1 decimal place.

$\theta$	0 ${}^{\circ }$	30 ${}^{\circ }$	60 ${}^{\circ }$	90 ${}^{\circ }$	120 ${}^{\circ }$	150 ${}^{\circ }$
$sin\theta$
$\theta$	180 ${}^{\circ }$	210 ${}^{\circ }$	240 ${}^{\circ }$	270 ${}^{\circ }$	300 ${}^{\circ }$	330 ${}^{\circ }$	360 ${}^{\circ }$
$sin\theta$

Let us look back at our values for $sin\theta$

As you can see, the function $sin\theta$ has a value of 0 at $\theta =0^{\circ }$ . Its value then smoothly increases until $\theta ={90}^{\circ }$ when its value is 1. We also know that it later decreases to 0 when $\theta ={180}^{\circ }$ . Putting all this together we can start to picture the full extent of the sine graph. The sine graph is shown in [link] . Notice the wave shape, with each wave having a length of ${360}^{\circ }$ . We say the graph has a period of ${360}^{\circ }$ . The height of the wave above (or below) the $x$ -axis is called the wave's amplitude . Thus the maximum amplitude of the sine-wave is 1, and its minimum amplitude is -1.

Functions of the form $y=asin\left(x\right)+q$

In the equation, $y=asin\left(x\right)+q$ , $a$ and $q$ are constants and have different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=2sin\theta +3$ .

Functions of the form $y=asin\left(\theta \right)+q$ :

1. On the same set of axes, plot the following graphs:
   1. $a\left(\theta \right)=sin\theta -2$
   2. $b\left(\theta \right)=sin\theta -1$
   3. $c\left(\theta \right)=sin\theta$
   4. $d\left(\theta \right)=sin\theta +1$
   5. $e\left(\theta \right)=sin\theta +2$
   Use your results to deduce the effect of $q$ .
2. On the same set of axes, plot the following graphs:
   1. $f\left(\theta \right)=-2·sin\theta$
   2. $g\left(\theta \right)=-1·sin\theta$
   3. $h\left(\theta \right)=0·sin\theta$
   4. $j\left(\theta \right)=1·sin\theta$
   5. $k\left(\theta \right)=2·sin\theta$
   Use your results to deduce the effect of $a$ .

You should have found that the value of $a$ affects the height of the peaks of the graph. As the magnitude of $a$ increases, the peaks get higher. As it decreases, the peaks get lower.

$q$ is called the vertical shift . If $q=2$ , then the whole sine graph shifts up 2 units. If $q=-1$ , the whole sine graph shifts down 1 unit.

These different properties are summarised in [link] .

	$a>0$	$a<0$
$q>0$
$q<0$

Domain and range

For $f\left(\theta \right)=asin\left(\theta \right)+q$ , the domain is $\{\theta :\theta \in \mathbb{R}\}$ because there is no value of $\theta \in \mathbb{R}$ for which $f\left(\theta \right)$ is undefined.

The range of $f\left(\theta \right)=asin\theta +q$ depends on whether the value for $a$ is positive or negative. We will consider these two cases separately.

If $a>0$ we have:

This tells us that for all values of $\theta$ , $f\left(\theta \right)$ is always between $-a+q$ and $a+q$ . Therefore if $a>0$ , the range of $f\left(\theta \right)=asin\theta +q$ is $\left\{f\right(\theta ):f(\theta )\in [-a+q,a+q\left]\right\}$ .

Similarly, it can be shown that if $a<0$ , the range of $f\left(\theta \right)=asin\theta +q$ is $\left\{f\right(\theta ):f(\theta )\in [a+q,-a+q\left]\right\}$ . This is left as an exercise.

Intercepts

The $y$ -intercept, $y_{int}$ , of $f\left(\theta \right)=asin\left(x\right)+q$ is simply the value of $f\left(\theta \right)$ at $\theta =0^{\circ }$ .

Graph of $cos\theta$

Graph of $cos\theta$ :

Complete the following table, using your calculator to calculate the values correct to 1 decimal place. Then plot the values with $cos\theta$ on the $y$ -axis and $\theta$ on the $x$ -axis.