Cosine graph

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# Cosine graph

To create this article, volunteer authors worked to edit and improve it over time. This article has been viewed 11, times. Learn more The sine and cosine functions appear all over math in trigonometry, pre-calculus, and even calculus. Understanding how to create and draw these functions is essential to these classes, and to nearly anyone working in a scientific field.

This article will teach you how to graph the sine and cosine functions by hand, and how each variable in the standard equations transform the shape, size, and direction of the graphs.

### Key Concepts

Create an account. Edit this Article. We use cookies to make wikiHow great. By using our site, you agree to our cookie policy. Learn why people trust wikiHow. Explore this Article parts. Tips and Warnings. Things You'll Need. Related Articles. Author Info Last Updated: February 6, Part 1 of Draw a coordinate plane. Since you are only hand-drawing your graphs, there is no precise scale, but it must be accurate at certain points. It may be helpful to use two separate colors to distinguish between sine and cosine.

Part 2 of Use the standard equation to define your variables. Calculate the period. Calculate the amplitude. Calculate the phase shift. Calculate the vertical shift. Graph the final function.By Mary Jane Sterling. The trigonometry equation that represents this relationship is.

Look at the graphs of the sine and cosine functions on the same coordinate axes, as shown in the following figure. The graphs of the sine and cosine functions illustrate a property that exists for several pairings of the different trig functions.

The property represented here is based on the right triangle and the two acute or complementary angles in a right triangle. The identities that arise from the triangle are called the cofunction identities. These identities show how the function values of the complementary angles in a right triangle are related. This equation is a roundabout way of explaining why the graphs of sine and cosine are different by just a slide. You probably noticed that these cofunction identities all use the difference of angles, but the slide of the sine function to the left was a sum.

The shifted sine graph and the cosine graph are really equivalent — they become graphs of the same set of points. You want to show that the sine function, slid 90 degrees to the left, is equal to the cosine function:.

Apply the two identities for the sine of the sum and difference of two angles. She has been teaching mathematics at Bradley University in Peoria, Illinois, for more than 30 years and has loved working with future business executives, physical therapists, teachers, and many others. Comparing Cosine and Sine Functions in a Graph.To graph the cosine function, we mark the angle along the horizontal x axis, and for each angle, we put the cosine of that angle on the vertical y-axis.

Curves that follow this shape are called 'sinusoidal' after the name of the sine function whose shape it resembles. In the diagram above, drag the point A around in a circular path to vary the angle CAB. As you do so, the point on the graph moves to correspond with the angle and its cosine. If you check the "progressive mode" box, the curve will be drawn as you move the point A instead of tracing the existing curve.

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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'. You can rotate the point as many times as you like. This means you can find the cosine of any angle, no matter how large.

In mathematical terms we say the 'domain' of the cosine function is the set of all real numbers. The range of a function is the set of result values it can produce.

Looking at the cosine curve you can see it never goes outside this range. What if we were asked to find the inverse cosine of a number, let's say 0. In other words: what angle has a cosine of 0. If we look at the curve above we see four angles whose cosine is 0. In fact, since the graph goes on forever in both directions, there are an infinite number of angles that have a cosine of a 0. If you ask a calculator to find the arc cosine cos -1 or acos of a number, it cannot return an infinitely long list of angles, so by convention it returns just the first one.

But remember, there are many others. Home Contact About Subject Index. When the cosine of an angle is plotted against that angle measure, the result is a classic shape similar to the cosine curve. Try this Drag the vertex of the triangle and see how the cosine function graph varies with the angle.Next, plot these values and obtain the basic graphs of the sine and cosine function Figure 1.

Several additional terms and factors can be added to the sine and cosine functions, which modify their shapes. This also holds for the cosine function Figure 3. This also holds for the cosine function Figure 4. These two functions have minimum and maximum values as defined by the following formulas. What are the maximum and minimum values of the function?

### Trigonometric functions

This also holds for the cosine function. The phase shift is D. This is a positive number. It does not matter whether the shift is to the left if D is positive or to the right if D is negative. The sine function is odd, and the cosine function is even.

In other words. Example 3: What is the amplitude, period, phase shift, maximum, and minimum values of. It is important to understand the relationships between the sine and cosine functions and how phase shifts can alter their graphs.

Previous Periodic and Symmetric Functions. Next Graphs Other Trigonometric Functions. Removing book from your Reading List will also remove any bookmarked pages associated with this title. Are you sure you want to remove bookConfirmation and any corresponding bookmarks?

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My Preferences My Reading List. Graphs: Sine and Cosine. Multiple periods of the a sine function and b cosine function. Adam Bede has been added to your Reading List!In mathematicsthe trigonometric functions also called circular functionsangle functions or goniometric functions   are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

They are widely used in all sciences that are related to geometrysuch as navigationsolid mechanicscelestial mechanicsgeodesyand many others.

How To Find The Amplitude, Period, Phase Shift, and Midline Vertical Shift of a Sine Cosine Function

They are among the simplest periodic functionsand as such are also widely used for studying periodic phenomena, through Fourier analysis. The most widely used trigonometric functions are the sinethe cosineand the tangent.

Their reciprocals are respectively the cosecantthe secantand the cotangentwhich are less used in modern mathematics.

Each of these six trigonometric functions has a corresponding inverse function called inverse trigonometric functionand an equivalent in the hyperbolic functions as well.

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The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extending these definitions to functions whose domain is the whole projectively extended real linegeometrical definitions using the standard unit circle i. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex planeand the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lower case letter denotes an edge of the triangle and its length. More precisely, the six trigonometric functions are:  . In geometric applications, the argument of a trigonometric function is generally the measure of an angle.

For this purpose, any angular unit is convenient, and angles are most commonly measured in degrees particularly in elementary mathematics. When using trigonometric function in calculustheir argument is generally not an angle, but a real number. In this case, it is more suitable to express the argument of the trigonometric as the length of the arc of the unit circle —delimited by an angle with the center of the circle as vertex. Therefore, one uses the radian as angular unit: a radian is the angle that delimits an arc of length 1 on the unit circle.

A great advantage of radians is that they make many formulas much simpler to state, typically all formulas relative to derivatives and integrals. Because of that, it is often understood that when the angular unit is not explicitly specified, the arguments of trigonometric functions are always expressed in radians.

## Comparing Cosine and Sine Functions in a Graph

The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circlewhich is the circle of radius one centered at the origin O of this coordinate system. The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A.

That is. By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is. That is, the equalities. The same is true for the four other trigonometric functions. The algebraic expressions for the most important angles are as follows:. Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.

Such simple expressions generally do not exist for other angles which are rational multiples of a straight angle.By Yang Kuang, Elleyne Kase. The parent graph of cosine looks very similar to the sine function parent graph, but it has its own sparkling personality like fraternal twins. Cosine graphs follow the same basic pattern and have the same basic shape as sine graphs; the difference lies in the location of the maximums and minimums. If necessary, you can refer to the unit circle for the cosine values to start with.

Here are the steps:.

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Like with sine graphs, the domain of cosine is all real numbers, and its range is. Using your knowledge of the range for cosine from Step 1, you know the highest value that cos x can be is 1, which happens twice for cosine — once at an angle of 0 and once at an angle of.

The minimum value that cos x can be is —1, which occurs at an angle of. It repeats every 2-pi radians. Unlike the sine function, which has degree symmetry, cosine has y- axis symmetry. In other words, you can fold the graph in half at the y- axis and it matches exactly. For example. Even though the input sign changed, the output sign for cosine stayed the same, and it always does for any theta value and its opposite.

Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. How to Graph a Cosine Function. About the Book Author Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years.A sine waveor sinusoidis the graph of the sine function in trigonometry. A sinusoid is the name given to any curve that can be written in the form A and B are positive. Sinusoids are considered to be the general form of the sine function. In addition to mathematics, sinusoidal functions occur in other fields of study such as science and engineering.

This function also occurs in nature as seen in ocean waves, sound waves and light waves. Even average daily temperatures for each day of the year resemble this function. The term sinusoid was first used by Scotsman Stuart Kenny in while observing the growth and harvest of soybeans. Any cosine function can be written as a sine function. The value A in front of sin or cos affects the amplitude height. The amplitude half the distance between the maximum and minimum values of the function will be Asince distance is always positive.

Increasing or decreasing the value of A will vertically stretch or shrink the graph. The value B affects the period. The period of sine and cosine is. This problem is a combination of dealing with the values of A and B. The A value of 3 tells us that the graph will have a vertical stretch and the amplitude will be 3. Look out for this problem. The amplitude is 2 a positive value representing distance. The amplitude of 2 tells us that the graph will have a vertical stretch.

Sinusoidal Graphs MathBitsNotebook. The number, Ain front of sine or cosine changes the height of the graph. Notice: The graphs change "height" but do not change horizontal width. Notice: These graphs change horizontal "width" but do not change height.

How to use your graphing calcualtor for graphing sinusoidal functions. NOTE: The re-posting of materials in part or whole from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the " Terms of Use ".  