Waves are everywhere. Lights, sounds, energy and more are all made up of waves. A wave is a periodic disturbance that moves energy between two places. Electromagnetic waves are used for cooking food in a microwave. Mechanical waves are found in the waves of the ocean. Sine waves and cosine waves are periodic functions, meaning the functions repeat themselves again and again over a consistent length of values. When referring to sine and cosine waves, the maximum and minimum heights of each wave are constant. Compare sine and cosine to the graph shown of the sound wave. Sound waves can have varying heights for each wave, and they are not necessarily periodic.
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Try it nowThe trigonometric function sine is defined as {eq}y=sin(x) {/eq}. The function oscillates around the line y = 0. This means the function goes up and down around the x-axis repeatedly. A sine wave, also called a sinusoidal wave, is represented by the sine function and is a smooth curve that repeats itself periodically.
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Sine waves are used to model musical tones, ocean tides, how far away one planet is from another, and stockbrokers even use sine waves to predict the best time to sell or buy a stock.
The reason sine waves garner such important status in trigonometry is that so many real-life applications (as mentioned in the previous paragraph) can be modeled with sinusoidal waves but not with any other trigonometric wave. And while these sine waves may not always match the situation perfectly, they do a much better job approximating the wave than most other functions ever could.
Listed below is some common terminology when talking about sine waves.
A sine wave is represented by the following sine function: {eq}y=asin(bx-c) {/eq} where
The basic sine function is {eq}y=sin(x) {/eq} where a = 1, b = 1, c = 0 and d = 0. Thus, the amplitude of this function is 1, the period is {eq}\dfrac{2\pi}{1} = 2\pi {/eq}, and the phase shift is {eq}\dfrac{0}{1}=0 {/eq}.
Five important points on the graph of {eq}sin(x) {/eq} are {eq}(0,0), (\dfrac{\pi}{2}, 1), (\pi, 0), (\dfrac{3\pi}{2}, -1), (2\pi, 0) {/eq}
These points are the locations of the maximum y-value, the minimum y-value, and the three points where the function intersects the x-axis.
Careful attention to order is necessary to graph a sine wave correctly.
Step 1: Identify the amplitude, {eq}|a| {/eq}.
{eq}|a|=|3|=3 {/eq}, so the amplitude is 3. This means the maximum y-value is 3 and the minimum y-value is -3.
Step 2: The period is found with the formula {eq}\dfrac{2\pi}{|b|} {/eq}.
{eq}\dfrac{2\pi}{|2|} = \pi {/eq}. The function will repeat itself after {eq}\pi {/eq} units.
Step 3: The phase shift is {eq}\dfrac{c}{|b|} = \dfrac{\pi}{2} {/eq}. The graph of the function shifts to the right by {eq}\dfrac{\pi}{2} {/eq} units.
Step 4: Find the new x and y-values.
In the original {eq}y=sin(x) {/eq} graph, the five important x-values were {eq}0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2}, 2\pi. {/eq}
To find the x-values of this function after the phase shift and shrinking of the period, we need to solve the following equations where the argument of the sine function is set equal to the original x-values:
{eq}2x-\pi=0\\ 2x-\pi= \dfrac{\pi}{2}\\ 2x-\pi= \pi\\ 2x-\pi= \dfrac{3\pi}{2}\\\ 2x-\pi= 2\pi {/eq}
First, add {eq}\pi {/eq} to each side.
{eq}2x=\pi\\ 2x= \dfrac{3\pi}{2}\\ 2x= 2\pi\\ 2x= \dfrac{5\pi}{2}\\ 2x= 3\pi {/eq}
Then, divide each side of the equations by 2.
{eq}x=\dfrac{\pi}{2}\\ x= \dfrac{3\pi}{4}\\ x= \pi\\ x= \dfrac{5\pi}{4}\\ x= \dfrac{3\pi}{2} {/eq}
These five values are the new locations of the minimum y-value, maximum y-value, and the x-intercepts.
Finding the y-values is simpler: multiply every y value by the amplitude. The original y-values were 0, 1, 0, -1, 0 and, once multiplied by the amplitude of 3, become 0, 3, 0, -3, 0.
Thus the points on the graph are
{eq}(\dfrac{\pi}{2},0)\\ (\dfrac{3\pi}{4},3)\\ (\pi,0)\\ (\dfrac{5\pi}{4},-3)\\ (\dfrac{3\pi}{2},0) {/eq}
Step 5: Plot the new x-values and y-values.
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Step 6: Connect all the points with a smooth continuous line.
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That is the completed sine graph.
The cosine wave is the same as the sine wave but shifted {eq}\dfrac{\pi}{2} {/eq} units to the right. That is, {eq}sin(x)=cos(x-\dfrac{\pi}{2}) {/eq}. Because of this equality, everything that holds true for the sine wave holds true for the cosine wave. Since cosine is shifted to the right by {eq}\pi/2 {/eq} units, there is a new set of five points: {eq}(0, 1), (\dfrac{\pi}{2}, 0), (\pi, -1), (\dfrac{3\pi}{2}, 0), (2\pi, 1) {/eq}
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Recall the definitions from the sine wave function:
A Cosine wave is represented by the following cosine function: {eq}y=acos(bx-c) {/eq} where
This is the graph of the basic cos(x) function where a = 1, b = 1, and c = 0.
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The steps for graphing the cosine function are the same as those for graphing the sine function.
Step 1: Identify the amplitude, |a|.
a = |5| = 5.
Step 2: Find the period by calculating {eq}\dfrac{2\pi}{|b|} {/eq}.
{eq}\dfrac{2\pi}{\dfrac{\pi}{3}}={2\pi}*\dfrac{3}{\pi}=6 {/eq}.
Step 3: Identify the phase shift, {eq}\dfrac{c}{b} {/eq}.
{eq}\dfrac{c}{b}=\dfrac{\dfrac{\pi}{2}}{\dfrac{\pi}{3} }=\dfrac{\pi}{2}*\dfrac{3}{\pi}=\dfrac{3}{2} {/eq}.
Step 4: Find the new x-values and y-values.
To do this, set the argument of the cosine function equal to each of the original five x-values on the graph of {eq}cos(x) {/eq}.
{eq}\dfrac{\pi}{3} x-\dfrac{\pi}{2} = 0\\ \dfrac{\pi}{3} x-\dfrac{\pi}{2}=\dfrac{\pi}{2}\\ \dfrac{\pi}{3} x-\dfrac{\pi}{2}=\pi\\ \dfrac{\pi}{3} x-\dfrac{\pi}{2}=\dfrac{3\pi}{2}\\ \dfrac{\pi}{3} x-\dfrac{\pi}{2}=2\pi {/eq}
Add {eq}\dfrac{\pi}{2} {/eq} to each side.
{eq}\dfrac{\pi}{3} x= \dfrac{\pi}{2}\\ \dfrac{\pi}{3} x=\pi\\ \dfrac{\pi}{3} x=\dfrac{3\pi}{2}\\ \dfrac{\pi}{3} x=2\pi\\ \dfrac{\pi}{3} x=\dfrac{5\pi}{2} {/eq}
Multiply every equation by 3.
{eq}\pi x= \dfrac{3\pi}{2}\\ \pi x=3\pi\\ \pi x=\dfrac{9\pi}{2}\\ \pi x=6\pi\\ \pi x=\dfrac{15\pi}{2} {/eq}
Finally, divide each equation by {eq}\pi {/eq}.
{eq}x= \dfrac{3}{2}\\ x=3\\ x=\dfrac{9}{2}\\ x=6\\ x=\dfrac{15}{2} {/eq}
To find the new y-values, multiply all the y-values by the amplitude, 5. Thus, the new points are
{eq}(\dfrac{3}{2}, 5), (3,0), (\dfrac{9}{2}, -5), (6, 0)(\dfrac{15}{2}, 5) {/eq}
Step 5 and Step 6: Plot the new x-values and y-values and connect all the points with a smooth continuous line.
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The amplitude of the sine wave is the distance from the x-axis to the maximum y-value. {eq}2|a| {/eq} is the total height of the wave from the minimum to the maximum y-value. Consider a Ferris Wheel that is 250 ft tall. Then the amplitude would be 250/2 = 125 ft. Thus, when building the structure for the wheel, the center of the wheel where the spokes meet should be 125 ft tall.
The period of a sine function describes how long the function takes to complete one cycle. Going back to the Ferris Wheel, if the ride makes a complete turn every 10 minutes, the period of the function would be 10 minutes.
There aren't many differences between the sine wave and the cosine wave. Recall that {eq}sin(x)=cos(x-\dfrac{\pi}{2}) {/eq}. Hence the sine wave is a quarter of the period behind the cosine wave. Both trigonometric functions are useful for modeling examples of waves in the real world, though sometimes one function will have benefits over using the other function. For instance, in a situation where the wave starts at its maximum value, the cosine function would be best. If the initial y-value starts at zero, then the sine function would be best to model the event.
Here is another example of graphing a sine wave.
Example: Graph the sine wave given by {eq}y=sin(x-\pi) {/eq}.
Step 1: The amplitude is 1.
Step 2: The period is {eq}2\pi {/eq}.
Step 3: The phase shift is {eq}\pi {/eq}.
Step 4. Solve the following equations to find the new x-values.
{eq}x-\pi=0\\ x-\pi= \dfrac{\pi}{2}\\ x-\pi= \pi\\ x-\pi= \dfrac{3\pi}{2}\\ x-\pi= 2\pi {/eq}
Add {eq}\pi {/eq} to each side of the equations.
{eq}x=\pi\\ x= \dfrac{3\pi}{2}\\ x= 2\pi\\ x= \dfrac{5\pi}{2}\\ x= 3\pi {/eq}
Since the amplitude is 1, the y-values do not change. Thus the five points are:
{eq}(\pi, 0), (\dfrac{3\pi}{2}, 1), (2\pi, 0), (\dfrac{5\pi}{2}, -1), (3\pi, 0) {/eq}
Step 5 and Step 6: Plot and connect the points.
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Example: Graph the cosine wave {eq}y=2cos(x) {/eq}
Step 1: The amplitude is 2.
Step 2: The period is {eq}2\pi {/eq}
Step 3: The phase shift is zero.
Step 4: Since the phase shift and period did not change from {eq}cos(x) {/eq}, the x-values will not change. The y-values will be multiplied by a factor of 2. Thus, the five points for this graph are: {eq}(0, 2), (\dfrac{\pi}{2}, 0), (\pi, -2), (\dfrac{3\pi}{2}, 0), (2\pi, 2), {/eq}
Step 5 and Step 6: Plot the points and connect them with a smooth continuous line.
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Waves move energy around using periodic disturbances. Sine waves and cosine waves are periodic functions, meaning the functions repeat themselves again and again over a consistent length of values. When referring to sine and cosine waves, the maximum and minimum heights of each wave are constant. These features make sine waves, and cosine waves good choices to model real-life applications of waves. Amplitude is the height of the function from the line y = 0 to its maximum y-value or minimum y-value. Both sine and cosine have an amplitude of 1 in their basic form. The period of the function is how long it takes the function to repeat itself. Sine and cosine both have natural periods of {eq}2\pi {/eq}. The phase shift is the number of units the function is shifted left or right. Finally, the frequency is how many times the wave repeats itself in an interval of length {eq}2\pi {/eq}.
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Sine waves are used to model waves occurring in the real world. This can range from uses in the stock market, to modeling the ocean tides, to examining shocks from earthquakes.
Identify the amplitude, period, phase shift, and the 5 main points. Carefully plot the points on a graph and then connect them with a smooth continuous curve.
An example of a cosine function is cos(x). Another example is 5cos(x). This vertically stretches the function cos(x) by 5 units. In the function cos(2x), the function is shrunk horizontally by a factor of 2. Finally, cos(x-3) shifts the graph of cos(x) 3 units to the right.
The sine function is an S-shaped curve that repeats itself every period of 2pi. The maximum y-value is 1 and the minimum y-value is -1.
Sine waves are different than cosine waves in that the sine wave is a quarter cycle behind the cosine wave. If one were to shift the sine wave by pi/2 to the left, it is identical in every respect to the cosine wave.
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