# 8.1 Graphs of the sine and cosine functions  (Page 2/13)

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[link] shows that the cosine function is symmetric about the y -axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that

## Characteristics of sine and cosine functions

The sine and cosine functions have several distinct characteristics:

• They are periodic functions with a period of $\text{\hspace{0.17em}}2\pi .$
• The domain of each function is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)\text{\hspace{0.17em}}$ and the range is $\text{\hspace{0.17em}}\left[-1,1\right].$
• The graph of is symmetric about the origin, because it is an odd function.
• The graph of is symmetric about the $\text{\hspace{0.17em}}y\text{-}$ axis, because it is an even function.

## Investigating sinusoidal functions

As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function    is known as a sinusoidal function    . The general forms of sinusoidal functions are

## Determining the period of sinusoidal functions

Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.

In the general formula, $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ is related to the period by $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}|B|>1,\text{\hspace{0.17em}}$ then the period is less than $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ and the function undergoes a horizontal compression, whereas if $\text{\hspace{0.17em}}|B|<1,\text{\hspace{0.17em}}$ then the period is greater than $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ and the function undergoes a horizontal stretch. For example, $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x\right),\text{\hspace{0.17em}}$ $B=1,\text{\hspace{0.17em}}$ so the period is $\text{\hspace{0.17em}}2\pi ,\text{}$ which we knew. If $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(2x\right),\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}B=2,\text{\hspace{0.17em}}$ so the period is $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ and the graph is compressed. If $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(\frac{x}{2}\right),\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}B=\frac{1}{2},\text{\hspace{0.17em}}$ so the period is $\text{\hspace{0.17em}}4\pi \text{\hspace{0.17em}}$ and the graph is stretched. Notice in [link] how the period is indirectly related to $\text{\hspace{0.17em}}|B|.$

## Period of sinusoidal functions

If we let $\text{\hspace{0.17em}}C=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}D=0\text{\hspace{0.17em}}$ in the general form equations of the sine and cosine functions, we obtain the forms

$y=A\mathrm{sin}\left(Bx\right)$
$y=A\mathrm{cos}\left(Bx\right)$

The period is $\text{\hspace{0.17em}}\frac{2\pi }{|B|}.$

## Identifying the period of a sine or cosine function

Determine the period of the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(\frac{\pi }{6}x\right).$

Let’s begin by comparing the equation to the general form $\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx\right).$

In the given equation, $\text{\hspace{0.17em}}B=\frac{\pi }{6},\text{\hspace{0.17em}}$ so the period will be

Determine the period of the function $\text{\hspace{0.17em}}g\left(x\right)=\mathrm{cos}\left(\frac{x}{3}\right).$

$\text{\hspace{0.17em}}6\pi \text{\hspace{0.17em}}$

## Determining amplitude

Returning to the general formula for a sinusoidal function, we have analyzed how the variable $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ relates to the period. Now let’s turn to the variable $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ so we can analyze how it is related to the amplitude , or greatest distance from rest. $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ represents the vertical stretch factor, and its absolute value $\text{\hspace{0.17em}}|A|\text{\hspace{0.17em}}$ is the amplitude. The local maxima will be a distance $\text{\hspace{0.17em}}|A|\text{\hspace{0.17em}}$ above the vertical midline of the graph, which is the line $\text{\hspace{0.17em}}x=D;\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}D=0\text{\hspace{0.17em}}$ in this case, the midline is the x -axis. The local minima will be the same distance below the midline. If $\text{\hspace{0.17em}}|A|>1,\text{\hspace{0.17em}}$ the function is stretched. For example, the amplitude of $\text{\hspace{0.17em}}f\left(x\right)=4\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is twice the amplitude of $\text{\hspace{0.17em}}f\left(x\right)=2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}|A|<1,\text{\hspace{0.17em}}$ the function is compressed. [link] compares several sine functions with different amplitudes.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Sin(A+B) = sinBcosA+cosBsinA
Prove it
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