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Chapter 4: Trigonometric Functions

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Assignments and Homework
Pre-Assessment activity. Introduce periodicity and periodic functions.

Graph the length of day for Seattle.  Use full sheet of graph paper.

Journal Reflection: Note that the “period” of sine is 360°.  What is the fraction of 360 that represents the period for the data in Tables 4.1, 4.2, and 4.3?
Graph the length of day from Table 4.2 (Pg. 249) on the same graph paper as today's Seattle data.   Identify their similarities and differences.  Explain, in simple language, why the graphs of day length had the noted similarities and differences.  Speculate how the graph would differ if the location were changed to Mexico City and Nome (Alaska).
Graph one cycle of y = sin(x).  What is a reasonable window for this graph?  If the general formula for a sine function is y = A•sin[B(x – C)] + D then give the values for A, B, C, and D.  Note: for now, set MODE to FUNC and DEGREE. Debrief homework.

Periodic Functions
(Pgs. 249-253).  Work Pg. 254 #1.

Note that we will use the following as the "general form" for a sine function used to model data
y = A•sin[B(x – C)] + D.

Journal Reflection: Summarize what "amplitude" and "period" mean based upon the examples in today's activity.
Pg. 254 #2.
Play the animation in the link.  In your homework notebook, explain (in detail!) the relationship between the position of the green dot and the red graph. Trigification I.

Note that we will use the following as the "general form" for a sine function used to model data
y = A•sin[B(x – C)] + D.

Journal Reflection:  Explain in simple language what each part of the general form of a sine function (A, B, C, and D) does to y = sin(x) to trace a given graph.
Determine the equation for Seattle Sunrise, Seattle Day Length, and Boston Day Length (e.g. using the general form of a sine function y = A•sin[B(x – C)] + D wherein you determine A, B, C, and D).  Show how you derive each part.

Note: test correction session tomorrow (4 December) after school.
Write a step-by-step procedure to create a graph (on paper, calculator not allowed) from the equation of a periodic function.

See “How To Sine” for step by step instructions for how to use a function to produce a simple, periodic sine graph.
Trigophone activity.

Journal Reflection:  Identify your favorite radio station. Read the article “Radio - Frequency” about radio waves. Calculate the wave length of one cycle.
Using the data from Table 4.4 (pg. 255):
  1. Plot the data for the two years in the data table.
  2.  Draw a smooth line that will approximate the data.
  3. Create an equation for the line drawn from part 2.
  4. Use the equation to predict the total electric consumption by bakeries in March 1990.
Note: test correction session TODAY (5 December) before school.
Plot the monthly heating cost of homeowners over a two year period from Table 4.3 on pg. 252.  Model the data with a sine function. Debrief homework and Trigophone.  Graph a sine wave from an equation.

Journal Reflection:  A sine wave is manipulated such that when the original function completes a full wave in 6.28 units, the manipulated wave completes a period in 1.57 units. In previous exercises, the original sine wave completed a cycle in 360 degrees. Explain the changes you would need to make to your equation to find the correct graph. Calculate the frequency of the manipulated wave.
The S&P 500 is an American stock market index based on 500 large companies that reflects the strength of the economy. The data in the table provides the S&P price-to-earnings ratios which adjusts the true stock price for inflation. Plot the values from the table from 1920 to 1960, curve to approximate the data, create an equation to model the data and check the equation in your calculator.

(Consider studying information in this link, optional)
Get ready for the quiz!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people. Quiz 4.1 (write a sine function that matches a given graph using degrees).  You may use  a 3 x 5 note card but may NOT use a calculator).

Journal Reflection:  Graph y = (7 – x)sin(5x) over 0 ≤ x ≤ 2π, 8 ≤ y ≤ 8 with MODE in RADIAN.  Explain why the graph behaves that way.
Write sine functions that model Pg. 286 #13 (Table 4.8) using radians.  Remember to show HOW you derived A, B, C, and D.
Write a formula to convert a given number of degrees to radians and another formula to convert a given number of radians to degrees. Oscillating Phenomena and Periodic Functions
  • Can You Do The Can-Can activity.
  • Lesson 4.2 (Pgs. 262 - 268 starting with "Radian Measure").
  • Begin Homework Problems.
Journal Reflection:  Explain how would you convert your height data from "Can You Do The Can Can" activity to match values produced by the graph of the sine function in radians (that is, an axis of oscillation at 0, amplitude of 1, a period of 2π and horizontal shift of 0).
Pgs. 269-270 # 2, 3.  Omit 2 b).
Explain, in simple language, how to determine the circumference of a circular object and how to get the length of the arc that is made from a 30° central angle.  Assume you are given the object and cannot measure the diameter (or radius). Add another "Coefficients" column to Trigification I and complete using radians.  Remember to explain HOW you derived A, B, C, and D for each problem.  Consider downloading the file on the Radian-Degree Conversion protractor.

Journal Reflection:  How are the coefficients A, B, C, and D changed when modeling in radians rather than degrees?
Complete today's activity.
Explain in simple language how to convert the general form of a sine function written in degrees into an equivalent form in radians. Begin the review problems: Pgs. 298-300 #1, 2, 5 (use algebra and assume radians), 6, 8, 9.

Journal Reflection: Explain why merely "connecting points" is inappropriate when graphing data.
Finish the assigned review problems.

Consider beginning the Chapter 4 Take-home (due Friday, 19 Dec).
Let x be an angle measure that is a multiple of ten between 0° and 90°.  SHOW (in a table) the calculations for each value of x of [sin(x)]2 + [cos(x)]2. In-class review.

Journal Reflection: For any angle measure, x, explain geometrically why  [sin(x)]2 + [cos(x)]2 = 1.  Give a non-trivial example (select an angle measure and compute).
Prepare for tomorrow's test.  Continue working on the Chapter 4 Take-home (due Friday, 19 Dec).
Get ready for the test!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people. Test 4.1.  You may use YOUR calculator and a 3 x 5 note card* (writing on both sides is permitted).  Measurement devices (e.g. ruler and protractor) are also allowed.  Expect questions on
  • Model a data set with periodic functions (could be in either degrees or radians).
  • Convert between radians and degrees.
*Note cards may not be mechanically reproduced (no photo copies, word processing, etc.).

Journal Reflection:  Complete the Google Reflection online.
Finish the Chapter 4 Take-home!
Get ready for the game!  Move to a seat where you have ample room, have your calculator & homework notebook ready MATHO, you will be provided a BINGO style card.

Journal Reflection:  What will you do over the break to keep current on your mathematical skills?
None.  :-)