How to find the period of a cosine function

how to find the period of a cosine function

Trigonometric functions

The smallest such value is the period. The basic sine and cosine functions have a period of 2?. The function sin x is odd, so its graph is symmetric about the origin. The function cos x is even, so its graph is symmetric about the y-axis. The graph of a sinusoidal function has the same general shape as a sine or cosine function. Cosine Function The cosine function is a periodic function which is very important in trigonometry. The simplest way to understand the cosine function is to use the unit circle. For a given angle measure ?, draw a unit circle on the coordinate plane and draw the angle centered at the origin, with one side as the positive x -axis.

This document derives the Fourier Series coefficients for several functions. The functions shown here are fairly simple, but the concepts extend to more complex functions. Consider the periodic pulse function shown below. It how to find the period of a cosine function an even function with period T.

The function is a pulse function with amplitude Aand pulse width T p. The function can be defined over one period centered around the origin as:. During one period centered around the origin. This can be a bit hard to understand at first, but consider how to find the period of a cosine function sine function.

The values for a n are given in the table below. Note: this example was used on the page introducing the Fourier Series. You can change n by clicking the buttons. As beforenote:. The values for a n are given in the table below note: this example was used on the previous page. Note that because this example is similar to the previous one, the coefficients are similar, but they are no longer equal to zero for n even.

In problems with even and odd functions, we can exploit the inherent symmetry to simplify the integral. We will exploit other symmetries later. Consider the problem above. We can then use the fact that for an even function, e t. This will often be simpler to evaluate than the original integral because one of the limits of integration is zero. Fubction, again, the pulse function.

We can also represent x T t by the Exponential Fourier Funcction. The average value i. Note: this is similar, but not identical, to the triangle wave seen earlier.

Note: this is similar, but not identical, to the sawtooth wave seen earlier. So far, all of the functions considered have been either even or odd, but most functions are neither. This presents no conceptual difficult, but may require more integrations. For example if the function x T t looks like the one below. Since this has no obvious symmetries, a simple Sine or Cosine Series does not suffice. For the Trigonometric Fourier Series, this requires three integrals.

For this reason, among others, the Exponential Fourier Series is often easier to work with, though what is considered white meat on a chicken lacks the straightforward visualization afforded by the Trigonometric Fourier Series.

From the relationship between the Trigonometric and Exponential Fourier Series. What continent did amerigo vespucci discover the function x T t has certain symmetries, we can simplify the calculation of the coefficients.

In other words, if you shift the function by half of a period, then the resulting function is the opposite the original function. The triangle wave has half-wave symmetry. See below for clarification.

The first two symmetries are were discussed previously in the discussions of the pulse function x T t is pfriod and the sawtooth wave x T t is odd. The reason the coefficients of the even harmonics are how to play low on the piano can be understood in the context of the diagram below.

The top graph shows a function, x T t with half-wave symmetry along with the first four harmonics of the Fourier Series only sines are what are painted aluminum wheels because x T t is odd. The bottom graph shows the harmonics multiplied by x T t. The odd terms from dunction 1st red and 3rd magenta harmonics will what is a ballet dress called a positive result because they are above zero more than fid are below zero.

The even terms green and cyan will integrate to zero because they are equally above and below zero. Though this is a simple example, the concept applies for more complicated functions, and for higher harmonics. The only funct ion discussed with half-wave symmetry was the triangle wave and indeed the coefficients with even indices are equal to zero as are all of the b n terms because of the even symmetry.

In that case the a 0 term would be zero and we have already shown that all the terms with even indices are zero, as expected. Simplifications can also be made based on quarter-wave symmetrybut these are not discussed here. A periodic function has quarter wave symmetry if it has half wave symmetry and it is either even or odd around its two half-cycles.

Since the coefficients c n of the Exponential Fourier Series are related to the Trigonometric Series by. However, in addition, the coefficients of c n contain some symmetries of their own. In particular. Since the function is even, we expect the coefficients of the Exponential Fourier Series to be real and even from symmetry properties.

Furthermore, we have already calculated the coefficients of the Trigonometric Seriesand could easily calculate those of the Exponential Series. Thf, let us do it cosone first principles. The Exponential Fourier Series coefficients are given by.

The last step in the how to find the period of a cosine function is performed so we can use the sinc function pronounced like " sink ". This function comes up often in Fourier Analysis. The graph on the left shows the time domain function. If you hit the middle button, you will see a square wave with a duty cycle of 0. The graph on the right shown the values of c n vs n as how to find the period of a cosine function circles vs n the lower of cozine two horizontal axes; ignore the top axis for now.

There are several important features to note as T p is varied. As beforenote: As you add sine waves of increasingly higher frequency, cosins approximation improves. The addition of higher frequencies better approximates the rapid changes, or details, i.

Gibb's overshoot exists on either side of the discontinuity. Because of the symmetry of the waveform, only odd harmonics 1, 3, 5, Funchion reasons for this are discussed below The rightmost button shows the sum of all harmonics up to the 21st harmonic, fibd not all of the individual sinusoids are explicitly shown on the plot. In particular harmonics between 7 and 21 are not shown. Note: As you add sine waves of increasingly higher frequency, the approximation gets better and better, and these higher frequencies better approximate the details, i.

Conceptually, this occurs because the triangle wave looks much more like the 1st harmonic, so the contributions of the higher harmonics are less.

Even with only the 1st few harmonics we have a very good approximation to the original function. There is no discontinuity, so no Gibb's overshoot. As before, only odd harmonics 1, 3, 5, There is Gibb's overshoot caused by the discontinuities. A function can have half-wave symmetry without being either even or odd.

Even Pulse Function (Cosine Series)

To find the equation of sine waves given the graph, find the amplitude which is half the distance between the maximum and minimum. Next, find the period of the function which is the horizontal distance for the function to repeat. If the period is more than 2pi, B is a fraction; use the formula period=2pi/B to find . Nov 30,  · The period of the sine function is 2?, which means that the value of the function is the same every 2? units. The sine function, like cosine, tangent, cotangent, and many other trigonometric function, is a periodic function, which means it repeats its values on regular intervals, or "periods.". The domain of the cosine function. As you drag the point A around notice that after a full rotation about B, the graph shape repeats. The shape of the cosine curve is the same for each full rotation of the angle and so the function is called 'periodic'. The period of the function is ° or 2? radians.

Figure 1. Light can be separated into colors because of its wavelike properties. 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. 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? We can create a table of values and use them to sketch a graph. Figure lists some of the values for the sine function on a unit circle.

Plotting the points from the table and continuing along the x -axis gives the shape of the sine function. See Figure. Again, we can create a table of values and use them to sketch a graph. Figure lists some of the values for the cosine function on a unit circle. As with the sine function, we can plots points to create a graph of the cosine function as in Figure. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers.

Figure shows several periods of the sine and cosine functions. Looking again at the sine and cosine functions on a domain centered at the y -axis helps reveal symmetries. As we can see in Figure , the sine function is symmetric about the origin. Figure shows that the cosine function is symmetric about the y -axis. Again, we determined that the cosine function is an even function.

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.

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. The local minima will be the same distance below the midline. Figure compares several sine functions with different amplitudes. We must pay attention to the sign in the equation for the general form of a sinusoidal function.

Determine the formula for the cosine function in Figure. To determine the equation, we need to identify each value in the general form of a sinusoidal function. The greatest distance above and below the midline is the amplitude. The maxima are 0. Determine the formula for the sine function in Figure. Determine the equation for the sinusoidal function in Figure. We could write this as any one of the following:.

While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes. Again, these functions are equivalent, so both yield the same graph. Try It Write a formula for the function graphed in Figure. Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.

Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph. Figure A horizontally compressed, vertically stretched, and horizontally shifted sinusoid. Then graph the function. Begin by comparing the equation to the general form and use the steps outlined in Figure. Figure shows one cycle of the graph of the function. We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.

A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the y -coordinate of the point as a function of the angle of rotation. Finding the Vertical Component of Circular Motion A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P , as shown in Figure.

Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in Figure. Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other.

A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.

Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example. Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4.

Putting these transformations together, we find that. A weight is attached to a spring that is then hung from a board, as shown in Figure.

It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. With a diameter of m, the wheel has a radius of The height will oscillate with amplitude The midline of the oscillation will be at The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.

Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.

Access these online resources for additional instruction and practice with graphs of sine and cosine functions.

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. Round answers to two decimal places if necessary. Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure.

The graph appears linear. The graph is symmetric with respect to the y -axis and there is no amplitude because the function is not periodic. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The wheel completes 1 full revolution in 10 minutes. Privacy Policy. Skip to main content. Periodic Functions.

Search for:. Use phase shifts of sine and cosine curves. 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. Figure 2. The sine function. Figure 3. Plotting values of the sine function. Figure 4. The cosine function. Figure 5. Figure 6. Odd symmetry of the sine function. Figure 7.



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