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In this section, you will:
  • Graph variations of  y=sin( x )  and  y=cos( x ).
  • Use phase shifts of sine and cosine curves.
A photo of a rainbow colored beam of light stretching across the floor.
Light can be separated into colors because of its wavelike properties. (credit: "wonderferret"/ Flickr)

White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.

Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions , we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.

Graphing sine and cosine functions

Recall that the sine and cosine functions relate real number values to the x - and y -coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function    . We can create a table of values and use them to sketch a graph. [link] lists some of the values for the sine function on a unit circle.

x 0 π 6 π 4 π 3 π 2 2 π 3 3 π 4 5 π 6 π
sin ( x ) 0 1 2 2 2 3 2 1 3 2 2 2 1 2 0

Plotting the points from the table and continuing along the x -axis gives the shape of the sine function. See [link] .

A graph of sin(x). Local maximum at (pi/2, 1). Local minimum at (3pi/2, -1). Period of 2pi.
The sine function

Notice how the sine values are positive between 0 and π , which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π and 2 π , which correspond to the values of the sine function in quadrants III and IV on the unit circle. See [link] .

A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.
Plotting values of the sine function

Now let’s take a similar look at the cosine function    . Again, we can create a table of values and use them to sketch a graph. [link] lists some of the values for the cosine function on a unit circle.

x 0 π 6 π 4 π 3 π 2 2 π 3 3 π 4 5 π 6 π
cos ( x ) 1 3 2 2 2 1 2 0 1 2 2 2 3 2 1

As with the sine function, we can plots points to create a graph of the cosine function as in [link] .

A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.
The cosine function

Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [ 1 , 1 ] .

In both graphs, the shape of the graph repeats after 2 π , which means the functions are periodic with a period of 2 π . A periodic function    is a function for which a specific horizontal shift    , P , results in a function equal to the original function: f ( x + P ) = f ( x ) for all values of x in the domain of f . When this occurs, we call the smallest such horizontal shift with P > 0 the period    of the function. [link] shows several periods of the sine and cosine functions.

Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.

Looking again at the sine and cosine functions on a domain centered at the y -axis helps reveal symmetries. As we can see in [link] , the sine function    is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because sin ( x ) = sin x . Now we can clearly see this property from the graph.

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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cm
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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
Krampah Reply
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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Mujahid
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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