Unit 3: Trigonometric Graphs
Goal: The student will demonstrate the ability to sketch and analyze trigonometric graphs and apply trigonometry to solve real-world problems.

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Objectives – The student will be able to:


a. Graph the sine, cosine, and tangent functions.

Mathematical Background/Clarifying Examples:
Connect to students' previous knowledge of finding input and output values to plot points on a coordinate plane. This conversation will extend into a discussion of periodic functions and their relationships. After graphing functions by hand, students can examine how to graph the trig functions in radian mode on the graphing calculator. This may be easier to use when comparing multiple function transformations quickly.


Example:
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Resources:

Website Link:
Sine and Cosine Graphs: This link shows the sine and cosine graphs, while tracking the values along the unit circle.
http://www.purplemath.com/modules/triggrph.htm



b. Identify the domain and range of a basic trigonometric function.

Mathematical Background/Clarifying Examples:
Have students construct appropriate domain and range intervals of the sine, cosine, and tangent curves. They will need to use the graphs to determine the x-values for domain and y-values for range. This should lead to a discussion about possible input and output values of the basic trigonometric functions.


Example:
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Resources:

Website Links:
Domain of the Sine Curve: This page shows the sine curve and states the domain and range.
http://www.analyzemath.com/Graphing/GraphSineFunction.html

Questions about Domain and Range: This page shows questions (with answers) about finding the domain and range of trigonometric functions.
http://www.analyzemath.com/GraphBasicTrigonometricFunctions/GraphBasicTrigonoFunction.html




c. Sketch transformations of the sine, cosine, and tangent graphs.

Mathematical Background/Clarifying Examples:
Allow students to explore changes to the a, b, c, and d values of a basic trigonometric function in standard form. The students may be able to construct their own concepts of how each value affects the function. After this exploration and discussion, be sure to clearly show students the values that affect amplitude, period, phase shifts, and vertical shifts.


Example:
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Clarifying Example Link:
Graphing Sine Example: This website is an example of graphing the sine curve with transformations.
http://www.wiziq.com/tutorial/18649-Graphing-a-sine-curve


Resources:

Website Links:
Amplitude of Sine and Cosine: This link shows a sine curve with amplitude = 50, as well as allows you to use a java applet. http://www.intmath.com/trigonometric-graphs/1-graphs-sine-cosine-amplitude.php

Period of Sine and Cosine: This link shows a sine curve with a change in the period, as well as shows a real-life example. http://www.intmath.com/trigonometric-graphs/2-graphs-sine-cosine-period.php

Graphing Examples and Explanation: This link shows sine and cosine graphs with transformations.
http://www.regentsprep.org/Regents/math/algtrig/ATT7/graphvocab.htm


Files:
Graphing Trig Functions: This file is a guided notes packet of instruction on how to graph the sine curve.


Graphing Practice with Shifts (degrees): This is a practice sheet for graphing with transformations, including shifts in degrees.




d. Sketch the cosecant, secant, and cotangent functions and their transformations.

Math Background/Clarifying Examples:
Have the students connect what they just learned about sine, cosine, and tangent graphs to the reciprocal functions. Students can create a table of values to plot the reciprocal functions. This discovery activity may lead to conclusions about transformations of the reciprocal functions, including amplitude, period, phase shifts, and vertical shifts. Also, be sure to highlight any input values that can not be included the domain of the reciprocal function, thus creating vertical asymptotes.

Example:
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Check out this website for illustrations of these curves.
http://www.regentsprep.org/Regents/math/algtrig/ATT7/othergraphs.htm


Resources:

Website Links:
Inverse trigonometric function graphs: This link shows a graph of the 3 basic trigonometric functions paired with their reciprocal functions. The best examples are located at the bottom of the webpage.
http://www.intmath.com/trigonometric-graphs/4-graphs-tangent-cotangent-secant-cosecant.php


File:
Graphing tangent and cotangent:This file provides a chart used to practice sketching values for the tangent and cotangent.




e. Identify and sketch the period, amplitude (if any), phase shift, zeroes, and vertical asymptotes (if any) of the six trigonometric functions.

Mathematical Background/Clarifying Examples:
Have students sketch a graph of all trigonometric functions, giving careful attention to zeroes and any possible vertical asymptotes. Take note that to "graph" implies for the student to create a table of values and plot critical values for a graph, whereas a "sketch" implies for students to put asymptotes and key features on a graph and then draw a curve containing these key features. A sketch is not a precise graph of the function.

Also have students use the graph of a trigonometric function to create an equation of the function. In other words, students will have to work backwards from the graph to the equation. They will need to find the a, b, c, and d values to put into the standard form of the appropriate trigonometric function.

Example: This document is an example of identifying and writing a sine equation from a given graph.



Resources:

Website Link:
Writing the Equation of the Graph: This link has a practice sheet of graphs for students to write the equation. The worksheet is under the heading "Amplitude and Period of Trigonometric Functions."
http://www.analyzemath.com/trigonometry_worksheets.html




f. Use trigonometric graphs to model and solve real-world problems.

Mathematical Background/Clarifying Examples:
Have students identify, use, and apply trigonometric graphs to model applied problems. While you may show students all 6 function graphs, you may want to emphasize sine, cosine, and tangent graphs use in real-life examples. This could include using the sine regression for a periodic set of data, simple harmonic motion, etc...

Example:
The following table gives the average monthly temperature for Cleveland, Ohio for the years 1971-2000. Create a model for the data.
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This file is an application example of sine regression.



Resources:

Website Links:
Modeling Weather Data: This link provides a lesson plan for modeling weather data using the sine and cosine functions. http://www.uen.org/Lessonplan/preview.cgi?LPid=25928

Files: This file has data to be modeled using sine regression on the graphing calculator.


Graphing Practice with Regression: This file has practice problems for sketching curves, as well as a regression problem.


Website Link:
Simple Harmonic Motion: This link shows a few applications of the sine and cosine functions.
http://www.intmath.com/trigonometric-graphs/5-applications-trigonometric-graphs.php