# 5.5 Graphs of trig functions

 Page 1 / 3

## 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 ${}^{\circ }$ $cos\theta$

the art of managing the production, distribution and consumption.
what is economics
what is Open Market Operation
dominating middlemen men activities circumstances
what Equilibrium price
what is gap
mirwais
who is good with the indifference curve
Dexter
What is diseconomic
what are the types of goods
WARIDI
how can price determination be the central problem of micro economics
marginal cost formula
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
boniphace
Foday
ok
Foday
how can price determination be the central problem if micro economics
simon
formula of cross elasticity of demand
what is ceteris paribus
what is ceteris parabus
Priyanka
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
Abdullah
What is broker
scor
land is natural resources that is made by nature
scor
What is broker
scor
what is land
kafui
What is broker
scor
land is natural resources that is made by nature
scor
whats poppina nigga turn it up for a minute get it
what is this?
Philo
am from nigeria@ pilo
Frank
am from nigeria@ pilo
Frank
so
owusu
what is production possibility frontier
owusu
it's a summary of opportunity cost depicted on a curve.
okhiria
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
ok let me know some of the questions please.
Effah
ok am not wit some if den nw buh by tommorow I shall get Dem
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
Ukpen
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
Babaisa
okay
Humaira
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Faisal
Mannan
Wat d meaning of management
disaster management cycle
cooperate social responsibility
igwe
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
OLANIYI
Difference between extinct and extici spicies
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Got questions? Join the online conversation and get instant answers!