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CHAPTER 5 Sinusoidal Waves. rms Value
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will produce a RIGHT TRIANGLE having an angle , where  will always be the angle between A and the x-axis. This is illustrated in Fig. 74 for the case where the independent variable (the angle ) has a value such that A falls in the second quadrant.
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Let us call  the subsidiary angle. Note that, regardless of the quadrant that A falls in,  itself will always be a simple angle in the range of 08 to 908, which we ll de ne to always be a positive angle. Notice, however, that the x values and y values can be either positive or negative, depending upon the particular quadrant that A falls in (which, in turn, depends upon the value of the angle ). This combination, of positive values of  from  08 to  908 and the positive and negative x and y values, is now utilized to de ne the values of sin , cos , and tan , for any positive or negative value of  of any magnitude. This is done as follows. To begin, inspection of Fig. 74 shows that if A falls in the FIRST quadrant, x and y are BOTH POSITIVE, if A falls in the SECOND quadrant, x is NEGATIVE and y is POSITIVE, if A falls in the THIRD quadrant, x and y are BOTH NEGATIVE, if A falls in the FOURTH quadrant, x is POSITIVE and y is NEGATIVE. Next, given that A, the length of the radius vector, is always to be taken as positive, and also, recalling the basic de nitions of the trigonometric functions from section 5.2,
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we now de ne that
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CHAPTER 5 Sinusoidal Waves. rms Value
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y A x cos  cos  A y tan  tan  x sin  sin  73 74 75
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where  is a positive or negative angle of any value, and where  is the subsidiary angle as previously described. Since the signs of x and y depend upon the quadrant that A falls in, it follows that the signs ( or ) of the functions sin , cos , and tan  will also depend upon the quadrant that A falls in. This is illustrated in Figs. 75 through 78. The relations between the functions follow from these gures and from eqs. (73), (74), and (75), and are so important that they should all be committed to memory.
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Fig. 75
Fig. 76
Fig. 77
Fig. 78
Before we leave this section there are some relationships, between the trigonometric functions for  and , that should be derived. This can be done with aid of Figs. 79 and 80.
Fig. 79
Fig. 80
CHAPTER 5 Sinusoidal Waves. rms Value
From Fig. 79: sin  y=A sin  y=A cos  x=A cos  x=A tan  y=x tan  y=x From Fig. 80: sin  y=A sin  y=A cos  x=A cos  x=A tan  y= x y=x tan  y= x y=x To summarize: sin  sin  cos  cos  tan  tan  76 77 78 ' sin  sin  ' cos  cos  ' tan  tan  ' ' cos  cos  ' tan  tan  thus: thus: sin  sin 
Problem 65 Find the value of each of the following by making use only of the table of values given in section 5.2. (All angles are in degrees.) (a) cos 115 (b) sin 35 (c) tan 155 (d) sin 255 (e) cos 95 (f) tan 285 (g) sin 285 (h) sin 188
Thus, with aid of Figs. 75 through 78, the original table of values, given in section 5.2, can now be extended to cover the full range of values from  08 to  3608. A very short form of such a table follows, which let us now examine. To do this, let us apply, to the following table of values, the same graphical procedure that we used at the beginning of this section. That is, let us now apply to the table the SAME PROCEDURE that gave us Figs. 66 and 67. The results of doing this are shown in Figs. 81 and 82. Figure 81 shows, graphically, the behavior of the two sinusoidal functions, y sin  and y cos , over the complete range from  08 to  3608. Figure 82 shows the behavior of y tan  over the same range of values of . A brief discussion follows. Consider Fig. 81 rst. As the gure shows, the sinusoidal functions, sin  and cos , have a maximum value of 1 and a minimum value of 1. Note that the two waves are displaced from each other by 90 degrees; this is summarized by saying that the two waves are 90 degrees out of phase. Other than that, the two waveshapes are exactly the same.
8 0 15 30 45 60 75 90 105 120 135 150 165 180 sin  0.0000 0.2588 0.5000 0.7071 0.8660 0.9659 1.0000 0.9659 0.8660 0.7071 0.5000 0.2588 0.0000
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