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This section describes the graphs of trigonometric functions.
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.
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 $f\left(\theta \right)=2sin\theta +3$ .
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] .
$a>0$ | $a<0$ | |
$q>0$ | ||
$q<0$ |
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.
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 $\theta ={0}^{\circ}$ .
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.
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