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A graph of sin(x) that shows that sin(x) is an odd function due to the odd symmetry of the graph.
Odd symmetry of the sine function

[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 cos ( x ) = cos   x .

A graph of cos(x) that shows that cos(x) is an even function due to the even symmetry of the graph.
Even symmetry of the cosine function

Characteristics of sine and cosine functions

The sine and cosine functions have several distinct characteristics:

  • They are periodic functions with a period of 2 π .
  • The domain of each function is ( , ) and the range is [ 1 , 1 ] .
  • The graph of y = sin   x is symmetric about the origin, because it is an odd function.
  • The graph of y = cos   x is symmetric about the y - 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

y = A sin ( B x C ) + D               and y = A cos ( B x C ) + D

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, B is related to the period by P = 2 π | B | . If | B | > 1 , then the period is less than 2 π and the function undergoes a horizontal compression, whereas if | B | < 1 , then the period is greater than 2 π and the function undergoes a horizontal stretch. For example, f ( x ) = sin ( x ), B = 1, so the period is 2 π , which we knew. If f ( x ) = sin ( 2 x ) , then B = 2, so the period is π and the graph is compressed. If f ( x ) = sin ( x 2 ) , then B = 1 2 , so the period is 4 π and the graph is stretched. Notice in [link] how the period is indirectly related to | B | .

A graph with three items. The x-axis ranges from 0 to 2pi. The y-axis ranges from -1 to 1. The first item is the graph of sin(x) for one full period. The second is the graph of sin(2x) over two periods. The third is the graph of sin(x/2) for one half of a period.

Period of sinusoidal functions

If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms

y = A sin ( B x )
y = A cos ( B x )

The period is 2 π | B | .

Identifying the period of a sine or cosine function

Determine the period of the function f ( x ) = sin ( π 6 x ) .

Let’s begin by comparing the equation to the general form y = A sin ( B x ) .

In the given equation, B = π 6 , so the period will be

P = 2 π | B |    = 2 π π 6    = 2 π 6 π    = 12
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Determine the period of the function g ( x ) = cos ( x 3 ) .

6 π

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Determining amplitude

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

Questions & Answers

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Practice Key Terms 5

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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