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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
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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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