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by: Adam Crona
Adam Crona
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This 4 page Class Notes was uploaded by Adam Crona on Saturday September 12, 2015. The Class Notes belongs to Math 1B at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/201873/math-1b-university-of-california-irvine in Mathematics (M) at University of California - Irvine.

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Date Created: 09/12/15
Jim Lambers IVIath 1B Fall Quarter 2004 05 Lecture 7 Notes These notes correspond to Section 51 in the text The Wrapping Function Consider a circle of radius one with its center located at the point 00 This point is called the origin and is often denoted by 0 The trigonometric functions were originally defined so that a point P on this circle could be related to the angle that the line segment P makes with the Since then trigonometric functions have been used in a wide variety of applications that have nothing to do with angles since they are useful for describing any quantity that tends to oscillate or cycle through a particular range of values repeatedly Therefore we will define the trigonometric functions also known circular functions without using angles instead we will proceed by relating a point on the circle to the distance traveled along the circle starting from the point 10 that lies on the In this lecture we begin this presentation of trigonometric functions by defining a function called the wrapping function that describes this relationship Suppose that an infinitely long string corresponding to the real number line is wrapped around the circle of radius 1 and center 00 in the following manner First the point on the string corresponding to the number zero is placed at the point 10 on the circle Then the portion of the string corresponding to the positive real is wrapped around the circle infinitely many times in the counterclockwise direction while the portion corresponding to the negative real is wrapped around the circle in the clockwise direction This wrapping of the string around the circle associates each real number with a point on the circle Specifically suppose that we start at 1 0 and travel a distance 1 along the wrapped string If we arrive at a point 19 then the wrapping function associates the real number 1 with the point 1 y if we traveled counterclockwise whereas if we traveled clockwise then the real number d is associated with the point 1 g De nition of the Wrapping Function We now state a precise definition of the wrapping function First however we assign a name to the circle of radius 1 with center at the origin since this circle is of such importance De nition 1 Unit Circle The unit circle the circle that has center 00 and radius 1 Equiv alently the unit circle the set of all points 1g such that x2g2 1 1 Now we are ready to define the wrapping function De nition 2 Wrapping Function The wrapping function W the function de ned by 2 at b 2 where J any real number and 10 the point on the unit circle such that if a line segment of length and initial point 10 wrapped around the unit circle in the counterclockwise direction if J gt 0 and in the clockwise direction if J lt 0 then the terminal point of the line segment 10 It is important to recognize that unlike other functions that we will discuss in this course the range of the wrapping function is not a set of real numbers but a set of points in the plane Any set can be the domain or range of a function it does not have to be a set of numbers Exact Values of the Wrapping Function For some values of 1 we can determine the exact value of W 1 We begin by recalling that the formula for the circumference C of a circle of radius r is C 2771 so the circumference of the unit circle is Err It follows from this fact and the fact that W0 10 that W27r 10 well Intuitively a line segment of length 277 can be wrapped around the unit circle exactly once with the same starting and ending point Similarly if a segment of length 7139 is wrapped around the circle starting at the point W0 10 it will only make it halfway around the circle and end at Vbr 10 By the same token a segment of length 772 will only wrap around onefourth of the circle and a segment of length 3772 will wrap around threefourths of the circle It can be seen from Figure 1 that 1quot772 0 1 and llquot3rr2 0 1 By wrapping line segments around the circle in the clockwise direction we can determine exact values of WJ for certain negative values of Proceeding in the previous paragraph we find that W 27r 10 W rr 10 W rr2 0 1 and W 3rr2 01 Using a little algebra we can determine other values of WW The point u it on the unit circle that is the midpoint of the arc between 10 and 01 is equal to Wrr4 since this arc being onefourth of the circle has length 772 We can see from a graph of the unit circle that the J and t coordinates of Wrr4 must be equal that is u 1 Since the point u it is on the unit circle it satisfies the equation of the unit circle and therefore u2 12 1 It follows from these two equations that u 1 E We conclude that W xi xi 3 4 2 f 2 Using right triangles and the Pythagorean Theorem we can compute even more values of W 1 Suppose that we inscribe a hexagon in the unit circle whose sides are of equal length and place Exact values ofthe wrapping Jnction l l Wnl201 WTE 10 W010 W27t10 W37t1201 Figure 1 Values of the wrapping function WJ at J 0 rrE 7r 37r2 and Er two of the vertices of the hexagon at the points 10 and 10 It follows that the vertices of the hexagon are located at the points WO WrrES WOW3 Wrr W47r3 and llquot57r3 The hexagon is shown in Figure 2 Let WWW3 13 Then can be seen in Figure 2 1quot2773 ab It follows from the fact that the sides of the hexagon have equal length and the fact that the points WrrEi and WErrEi define a side of the hexagon that each side of the hexagon has length 2a Because 11 is on the unit circle we have a 12 1 Because the points 11 10 and 10 form a right triangle Whose hypotenuse is a side of the hexagon it follows from the Pythagorean Theorem that b2 1 a2 202 Solving these two equations for a and I yields a 12 and I xgE We conclude that 7r 1 xS l 3 i7 4 Using symmetry we can use our previous results to compute the value of the wrapping function at several more numbers We list these numbers and the corresponding points in the following Computing more exact values of the wrapping function 1 1 15 1 W27tl3ab W7r13ab 39 a a 05 r b 2a W0110 gt n Wow 10 a o5 r 1 W47r13ab W57t3ab 7 1 E 1 1 1 1 1 5 1 o 5 o o 5 1 1 5 Figure 2 Hexagon with equal sides inscribed within the unit circle The vertices are associated with the indicated numbers by the wrapping function HquotJ


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