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Module 5-II – Trigonometric Functions

Section 5-II.2 – Graphs of Trigonometric Functions

 

Introduction

Because of their periodicity, trigonometric functions are useful for modelling all kinds of periodic or cyclical phenomena. In order to match any particular oscillation, it may be necessary to shift or scale the basic sine graph we sketched in the last section. In this section you will develop more skill with these ideas, but the basic techniques of shifting and scaling, and graphing of sums and products should all be familiar to you from Module 2.

So don’t be intimidated by the presence of trig functions. There’s really not much new here.

 

Section Learning Objectives

 

·        Sketch graphs of trig functions and of others related to them by composition with linear functions

·        Identify equations from given graphs and from verbal descriptions of periodic phenomena

 

 

What to Do

1.         Read  section 5.5 of the text.

2.         Read the following Study Notes and Discussion, and make sure that you have absorbed the main points by answering the questions.

3.         Read section 5.6 of the text.

4.         Follow the instructions regarding Further Practice.

 

 Study Notes and Discussion

 

The breathing cycle and number of hours of daylight as discussed in the text’s Examples 11 and 12 in section 5.5 are just two of many examples of a quantity which varies up and down between certain bounds and which can be modelled by an equation of the form .  Several more are introduced in the exercises on pages 428-430, and another important example is the case of “harmonic motion” discussed in Example 6 of the text’s section 5.7 (on page 445).

 

The discussion you read in section 5.5 of the text has explored some of the properties of such graphs, but the material should not feel particularly new or unfamiliar to you as you could have done most of the examples back in Unit 2. Try using the shifting and scaling ideas of Unit 2 to describe how the graph of  is related to .(see answer #1)

 

In such situations, since the sine function oscillates between -1 and +1, the extremes of  are given by the constant term plus or minus the multiplier of the sine term(i.e. by  and  in the formula above). The average value is thus given by the constant term   , and the size of the multiplier  gives the magnitude of the variation away from that average. (This variation corresponds to the size of the waves and is called the amplitude).

 

The frequency and period of oscillation are related to the coefficient of t . In fact, for the sine function to go through a complete cycle, its argument must vary through an interval of length , and for  to change by , t must change by . So for the cycle to have length T, we must have . Or, going the other way, if we are given T then the coefficient of t is given by .

 

Finally, the constant, d, added inside the sine function depends on at what times the value of the quantity is equal to its average value, A. If  and , then  is passing its average A on the way up when . If    this happens at , but in general it happens at  so the graph is shifted left by .  Conversely, if we are told that  at, say, , then , so .

 

Note: The term phase shift is used in our text for the horizontal shift (or time shift when time is the independent variable)(  in the above discussion). This use of the term is common in precalculus texts but is not universal. In fact most science and engineering uses of the term apply it to the angular shift (just d in the above discussion), and you are unlikely to see it used in your calculus course. So it will not be used on the exam in this course. Whenever you do see the term phase shift in future, it would be safest to make sure to find out in what sense it is being used. In the meantime don’t worry about it.

 

Try to sketch the graph, and then find an equation of the form ,

for the function  described as follows:

 varies between 1 and 5 with a period of 6 and at  is increasing with a value of 2.

(see answer #2)

 

 

 

 

 

More Reading

 

Section 5.6 of the text applies the same shifting and scaling ideas from Unit 2 to the graphs of other trig functions, and also includes examples reminding you of the process of adding ordinates to graph the sum of two functions, and the use of intercepts and sign checking to help graph products.

 

 

 

 

Further Practice

Try enough of the odd numbered questions in the text sections 5.3 and 5.4, using the solutions guide if you need to, until you feel able to do similar questions on your own.

 

From section 5.5 you should do at least #1,11,31,41,51,and 61,

And from section 5.6 at least #15,35,55, and 65.