Graphing Sine and Cosine Functions

Sine and cosine can be interpreted in multiple ways, such as the relationship between angles and side lengths in triangles, and as the coordinates of points on the unit circle. Since sine and cosine return exactly one output value for every input angle, they can also be interpreted as functions. $y = sin (x) and y = cos (x) .$

Sine and cosine can be evaluated for any value of

x .

Thus, their domain is all real numbers. Every output, however, ranges from

- 1

to

1

— the range of both functions is

- 1 \leq y \leq 1 .

Concept

The Graphs of $sin (x)$ and $cos (x)$

The parent functions of sine and cosine functions are $y = sin (x)$ and $y = cos (x),$ respectively. Thus, it is important to know how their graphs look and some of their characteristics. Graphing them is done by first evaluating some inputs and obtaining their corresponding outputs using a table. Here, radians will be used, but the functions can be evaluated for degrees as well.

Plotting these points and connecting them with smooth curves gives the graphs.

$.$

Both functions have the amplitude

1,

which is half the difference between the greatest and least function values. These functions are both periodic, as they have a repeating pattern. The smallest repeating part of the graph is called a cycle. The horizontal length of the cycles is called the function's period, which is

2 π

for both

y = sin (x)

and

y = cos (x) .

Rule

Stretching and Shrinking Sine and Cosine Functions

To find more sine and cosine functions, the parent functions can be stretched or shrunk both horizontally and vertically, leading to the functions $y = a sin (b x) and y = a cos (b x) .$ The amplitude is affected by the factor $a .$ Notice though that an amplitude is always positive, while $a$ could be negative. Similarly, the period is affected by the constant $b .$ A larger $b$ shrinks the functions horizontally, leading to a shorter period. Here, as well, $b$ could be negative while a period is always positive. Thus, the amplitude and period of these functions are as follows.

Amplitude = $∣ a ∣$ Period = $\frac{2 π}{∣ b ∣}$

Find the amplitude and period of the function

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Exercise

The graph shows the function $y = 1.5 sin (1.6 x) .$

Using the graph, find the amplitude and period of the function. Then, verify them using the rule.

Show Solution

Solution

We can find the amplitude of the function by reading the greatest and least function values and calculating half the difference between them. From the graph, it can be seen that the greatest value is $1.5$ and the least is $- 1.5 .$

Calculating the difference between these values and dividing by two gives us $amplitude = \frac{1 . 5 - ( - 1 . 5 )}{2} = \frac{3}{2} = 1.5 .$ The period is found by identifying the horizontal length of a cycle. We'll look at the cycle starting at the origin, ending the next time the graph passes the $x$ -axis from below.

This cycle starts at $x = 0$ and ends at $x = 1.25 π .$ Thus, we can conclude that $period = 1.25 π .$ When a sine function is written in the form $y = a sin (b x),$ which the given function is, the amplitude is the absolute value of $a .$ Therefore, we find that $amplitude = ∣ 1.5 ∣ = 1.5 .$ For the period, we divide $2 π$ by the absolute value of $b .$ $period = \frac{2 π}{∣ 1 . 6 ∣} = \frac{2 π}{1 . 6} = 1.25 π$ Thus, the previous results have now been verified.

Rule

Translating Sine and Cosine Functions

By translating the functions $y = a sin (b x)$ and $y = a cos (b x),$ every sine and cosine function can be created. The resulting function rules are then $y = a sin (b (x - h)) + k and y = a cos (b (x - h)) + k,$

where

h

is the horizontal and

k

is the vertical translation. The horizontal line

y = k

is called the midline of the graph and is located in-between the

y

-values of the maximums and minimums. The graph's intersections with the midline corresponds to the

x

-intercepts of the parent function.

Method

Graphing Sine and Cosine Functions

Viewing sine and cosine functions as transformations of their parent functions, $y = sin (x)$ and $y = cos (x),$ makes it easier to graph these by hand. As an example, consider the function $f (x) = 0.5 cos (2 x) + 1 .$

1

Find the amplitude, period, and translation of the function

Identifying the amplitude, period, and translation allows the function to be viewed as the transformation that it is. For this function, the amplitude is $0.5,$ the period is $\frac{2 π}{∣ 2 ∣} = \frac{2 π}{2} = π,$ and it's translated $1$ unit upward.

2

Draw the midline

y = k

As the points intersecting the midline correspond to the $x$ -intercepts of the parent function, the midline should be the first thing graphed. Here, the midline is $y = 1 .$

3

Plot some key points on the graph

Key points ranging over at least one cycle of the function should now be plotted, as they can then be connected with a smooth curve to create the graph. These key points are the maximums, minimums, and intersections with the midline. Starting with the maximums, they occur at $x = 0, 2 π, 4 π, \dots$ for $y = cos (x),$ which is once every cycle. The period of $f$ is $π,$ and it's not been translated horizontally. Thus, the maximums of $f$ occur at $x = 0, π, 2 π, \dots$ As the midline is $y = 1$ and the amplitude is $0.5,$ the maximums will all have the $y$ -values $1 + 0.5 = 1.5 .$ They can now be plotted in the graph with the midline.

The minimums are located horizontally between the maximums, which is at $x = 0.5 π, 1.5 π, 2.5 π, \dots$ for $f .$ These have the $y$ -values $1 - 0.5 = 0.5 .$

Lastly, in-between every neighboring maximum and minimum are the intersections with the midline: $x = 0.25 π, 0.75 π, 1.25 π, \dots$ These points have the $y$ -value $1 .$

4

Draw the graph

The graph can now be drawn through the plotted points, continuing periodically in both directions.

Interpret the sine function

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Exercise

Riding a certain ferris wheel can be described using the function $h (t) = 100 sin (π (t - 0.5)) + 104,$ where $h$ is the height above ground in feet, and $t$ is the time since the ride started given in minutes.

Using the function rule, answer the following questions.

What is the diameter of the ferris wheel?
One ride on the ferris wheel is two full rotations. How long does a ride take?
How high above ground is the entry of the ferris wheel?

Show Solution

Solution

Example

Diameter of the ferris wheel

The diameter of the ferris wheel can be thought of as the difference in height between the highest and lowest point. Note that the amplitude is half this difference. Thus, the diameter is equal to twice the amplitude of the function. From the rule we find that the amplitude is $100,$ which gives us $diameter = 100 \cdot 2 = 200 ft,$ as the height is given in feet.

Example

Length of a ride

One full rotation of the wheel corresponds to one cycle of the function, as it will then repeat itself. Thus, one full rotation takes as long as one period of the function is. We can find the period as usually: $period = \frac{2 π}{∣ π ∣} = \frac{2 π}{π} = 2 min .$ One ride is twice this time, $4$ minutes.

Example

Entry's height above ground

Realistically, the ride should start at the same height as the entry. That is, when $t = 0 .$ Therefore, substituting $t = 0$ into the function gives us the height of the entry.

h (t) = 100 sin (π (t - 0.5)) + 104

Substitute

t = 0

h (0) = 100 sin (π (0 - 0.5)) + 104

SubTermSubtract term

h (0) = 100 sin (π (- 0.5)) + 104

MultiplyMultiply

h (0) = 100 sin (- 0.5 π) + 104

As the angle $- 0.5 π$ is equal to $- 9 0^{\circ},$ it corresponds to the coordinate $(0, - 1)$ on the unit circle. Thus, $sin (- 0.5 π)$ is equal to $- 1 .$

h (0) = 100 sin (- 0.5 π) + 104

Substitute

sin (- 0.5 π) = - 1

h (0) = 100 (- 1) + 104

MultiplyMultiply

h (0) = - 100 + 104

AddTermsAdd terms

h (0) = 4

We can now conclude that the entry to the ferris wheel is $4$ feet off ground level.