To create this article, volunteer authors worked to edit and improve it over time. This article has been viewed 9, 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. Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker.
Since you are only hand-drawing your graphs, there is no precise scale, but it must be accurate at certain points.Please ensure that your password is at least 8 characters and contains each of the following:.
Enter a problem Hope that helps! You'll be able to enter math problems once our session is over. Trigonometry Examples Step-by-Step Examples. Graphing Trigonometric Functions. Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift. Find the amplitude. Find the period of. The period of the function can be calculated using. Replace with in the formula for period. The absolute value is the distance between a number and zero.
The distance between and is. Cancel the common factor of. Cancel the common factor. Divide by. Find the phase shift using the formula. The phase shift of the function can be calculated from. Replace the values of and in the equation for phase shift.
Find the vertical shift. List the properties of the trigonometric function. Select a few points to graph. Find the point at. Replace the variable with in the expression.
Transformation of Sine and Cosine Graphs: Examples
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Sin And Cos Graph
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All rights reserved.By Sharon K. This is an exploration for Advanced Algebra or Precalculus teachers who have introduced their students to the basic sine and cosine graphs and now want their students to explore how changes to the equations affect the graphs. This is an introductory lesson whose purpose is to connect the language of Algebraic transformations to the more advanced topic of trignonometry. A key follows the end of the exploration. Consider the basic sine equation and graph.
If the first function is rewritten as…. Take a look at the blue and red graph and their equations. The graph of the first function remains in black. Equation of blue graph. Equation of red graph. Describe how the equation of the first function has changed to become the equation of the graphs in blue and red. This value is called the amplitude of the graph. Describe how the graph of the first function has changed to become the blue graph and the red graph.
Be as specific as possible.
Include in your answer how a specific point on the graph of the first function transforms to become a point on the blue graph and on the red graph. Take a look at an animation of this phenomenon.
Click here …. Be patient. These movies take awhile to load…. Now consider the purple and green graph and their equations. Equation of purple graph. Equation of green graph. Describe how the equation of the first function has changed to become the equation of the graphs in purple and green. Describe how the graph of the first function has changed to become the purple graph and the green graph.
Include in your answer how a specific point on the graph of the first function transforms to become a point on the purple graph and on the green graph. Look deeper than horizontal versus vertical…. Take a look at this phenomenon…. Examine the purple and green graph and their equations. Describe how the graph of the first function has changed to become the red graph and the green graph.
See how it moves…. Consider the blue and purple graph and their equations. This value is called the phase shift of the graph. Describe how the equation of the first function has changed to become the equation of the graphs in blue and purple.
Graphs of Trigonometric Functions Worksheet PDF
Describe how the graph of the first function has changed to become the blue graph and the purple graph. Include in your answer how a specific point on the graph of the first function transforms to become a point on the blue graph and on the purple graph. Take a look…. What about negatives? Examine the following graphs….Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials.
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This resource contains TWO seasonal activities! Use the snowflakes in the winter or the flowers in the spring -- depending on the pace of your course! In both versions, students will practice their knowledge of the properties of sine and cosine graphs.
The concepts assessed in this resource are: amp. TrigonometryWinterSpring. Add to cart. Wish List. Matching Sine and Cosine Graphs and Equations. Students will need to match one from each category together: 1.Each year my Pre-calculus students first encounter the graphs of sine and cosine curves, I find two groups of learners in my room.
I also found the spaghetti activity already created in paper and pencil format at NCTM Illuminations. You can download your copy HERE. To begin graphing and translating graphs for Sine and Cosine functions this year, I had my students complete this page in their interactive notebooks.
Both the sine function and the cosine function need 5-key points to complete one revolution. There is a starting point and a stopping point which divides the graph into four equal parts. I want them to learn the positive A-value patterns for the 5-key points of each function. Now, for the real test. When it comes time to translate these functions with phase shifts, horizontal and vertical transformations, reflections, along with all four values A, B, C, and D in the standard form equation, I want my students to develop some procedural techniques for themselves to better understand how easy these graphs can be sketched.
Patience and practice are the key. I have my own personal procedural techniques that I model for my students prior to asking them to create their own step-by-step approach. If so, sketch that line and label it x- prime. Once they get the five key points plotted, the last thing to do is to connect the points with a smooth curve and then repeat the pattern if you want them to sketch two full periods of the function.
Let me know what you think. Do you have any tips or tricks to graphing these sinusoidal curves? Now check your email to confirm your subscription. We are happy to have you join our Flamingo Math flock! Your email address will not be published. Notify me of follow-up comments by email. Notify me of new posts by email. This site uses Akismet to reduce spam. Learn how your comment data is processed.
Additional menu. You can download your copy HERE To begin graphing and translating graphs for Sine and Cosine functions this year, I had my students complete this page in their interactive notebooks. There was an error submitting your subscription. Please try again. Email Address. Share this: Tweet. ConvertKit Form. Leave a Reply Cancel reply Your email address will not be published.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 Functionswe examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions. 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. Plotting the points from the table and continuing along the x -axis gives the shape of the sine function.
Again, we can create a table of values and use them to sketch a graph. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. Now we can clearly see this property from the graph. 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.