# 5.5 Graphs of trig functions

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## Graphs of trigonometric functions

This section describes the graphs of trigonometric functions.

## 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 $sin\theta$

 $\theta$ ${0}^{\circ }$ ${30}^{\circ }$ ${45}^{\circ }$ ${60}^{\circ }$ ${90}^{\circ }$ ${180}^{\circ }$ $sin\theta$ 0 $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ 1 0

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 $\left\{\theta :\theta \in \mathbb{R}\right\}$ 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:

$\begin{array}{ccc}\hfill -1\le sin\theta & \le & 1\hfill \\ \hfill -a\le asin\theta & \le & a\hfill \\ \hfill -a+q\le asin\theta +q& \le & a+q\hfill \\ \hfill -a+q\le f\left(\theta \right)& \le & a+q\hfill \end{array}$

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 $\left\{f\left(\theta \right):f\left(\theta \right)\in \left[-a+q,a+q\right]\right\}$ .

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

The easiest way to find the range is simply to look for the "bottom" and the "top" of the graph.

## 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 $\theta ={0}^{\circ }$ .

$\begin{array}{ccc}\hfill {y}_{int}& =& f\left({0}^{\circ }\right)\hfill \\ & =& asin\left({0}^{\circ }\right)+q\hfill \\ & =& a\left(0\right)+q\hfill \\ & =& q\hfill \end{array}$

## 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.

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

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