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Unformatted text preview: Chapter 10 Trigonometric functions In this chapter we will explore periodic and oscillatory phenomena. The trigonometric functions will be the basis for much of what we construct 1 , and hence, we first introduce these and familiarize ourselves with their properties. 10.1 Introduction: angles and circles Angles can be measured in a number of ways. One way is to assign a value in degrees, with the convention that one complete revolution is represented by 360 ◦ . Why 360? And what is a degree exactly? Is this some universal measure that any intelligent being (say on Mars or elsewhere) would find appealing? Actually, 360 is a rather arbitrary convention that arose historically, and has no particular meaning. We could as easily have had mathematical ancestors that decided to divide circles into 1000 “equal pieces” or 240 or some other subdivision. It turns out that this measure is not particularly convenient, and we will replace it by a more universal quantity. The universal quantity stems from the fact that circles of all sizes have one common geometric feature: they have the same ratio of circumference to diameter, no matter what their size (or where in the universe they occur). We call that ratio π , that is π = Circumference of circle Diameter of circle The diameter D of a circle is just D = 2 r so this naturally leads to the familiar relationship of circumference, C , to radius, r , C = 2 πr (But we should not forget that this is merely a definition of the constant π . The more interesting conclusion that develops from this definition is that the area of the circle is A = πr 2 , but we shall see the reason for this later, in the context of areas and integration.) 1 Chapters 10 and 11 could be abbreviated and taught together to save some time at the end of the course. Indeed the details of the inverse trigonometric functions in Chapter 11 are not as essential as a basic understanding of the periodic functions. v.2005.1 - September 4, 2009 1 Math 102 Notes Chapter 10 θ s Figure 10.1: The angle θ in radians is related in a simple way to the radius R of the circle, and the length of the arc S shown. From Figure 10.1 we see that there is a correspondence between the angle ( θ ) subtended in a circle of given radius and the length of arc along the edge of the circle. For a circle of radius R and angle θ we will define the arclength, S by the relation S = Rθ where θ is measured in a convenient unit that we will now select. We now consider a circle of radius R = 1 (called a unit circle ) and denote by s a length of arc around the perimeter of this unit circle. In this case, the arc length is S = Rθ = θ We note that when S = 2 π , the arc consists of the entire perimeter of the circle. This leads us to define the unit called a radian : we will identify an angle of 2 π radians with one complete revolution around the circle. In other words, we use the length of the arc in the unit circle to assign a numerical value to the angle that it subtends.value to the angle that it subtends....
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- Winter '09
- Trigonometry, Sin, Cos