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CHAPTER 5 Sinusoidal Waves. rms Value
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the angle , the COSINE ( KOH sign ) of the angle , and the TANGENT of the angle . Then, The expression sine of  is abbreviated sin  and read as sine of theta, The expression cosine of  is abbreviated cos  and read as cosine of theta, The expression tangent of  is abbreviated tan  and read as tangent of theta. Referring now to the standard reference triangle of Fig. 64, the above expressions are de ned to mean that (and these de nitions should be committed to memory) sin  cos  tan  that is, in Fig. 64: sin  b=h cos  a=h tan  b=a 69 70 71 opposite side b hypotenuse h adjacent side a hypotenuse h opposite side b adjacent side a
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The quantities sin , cos , and tan  are the three principal trigonometric functions. The values of the functions depend only upon the angle . In our work the angle  will generally be regarded as the independent variable. As mentioned previously, sin  and cos  are referred to as the sinusoidal functions. Now let us see how the values of sin , cos , and tan  vary as  varies from  08 to  908. To help us to do this, we ve used our calculator to ll out a short four-place table of values as follows (which you can verify on your own calculator).
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8 0 1 2 3 4 5 6 8 10 15 20 25 30 sin  0.0000 0.0175 0.0349 0.0523 0.0698 0.0872 0.1045 0.1392 0.1737 0.2588 0.3420 0.4226 0.5000 cos  1.0000 0.9999 0.9994 0.9986 0.9976 0.9962 0.9945 0.9903 0.9848 0.9659 0.9397 0.9063 0.8660 tan  0.0000 0.0175 0.0349 0.0524 0.0699 0.0875 0.1051 0.1405 0.1763 0.2680 0.3640 0.4663 0.5774 8 35 40 45 50 55 60 65 70 75 80 85 88 90 sin  0.5736 0.6428 0.7071 0.7660 0.8192 0.8660 0.9063 0.9397 0.9659 0.9848 0.9962 0.9994 1.0000 cos  0.8192 0.7660 0.7071 0.6428 0.5736 0.5000 0.4226 0.3420 0.2588 0.1737 0.0872 0.0349 0.0000 tan  0.7002 0.8391 1.0000 1.1918 1.4281 1.7321 2.1445 2.7475 3.7321 5.6713 11.4301 28.6363 1
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A discussion of the table follows, in which it will be helpful to refer to Fig. 65.
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CHAPTER 5 Sinusoidal Waves. rms Value
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Fig. 65.
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Here we are holding h constant in length, as the angle h is allowed to have any value from 0 degrees to 90 degrees. As the angle h changes, the length of the sides a and b change, but h remains constant in length.
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Let us begin our discussion of the foregoing table for the special cases of  08 and  908, by making use of eqs. (69), (70), and (71), and Fig. 65, as follows. First, for  08 we have 9 sin  b=h 0=h 0; that is; sin 08 0 > = cos  a=h h=h 1; that is; cos 08 1 see table > ; tan  b=a 0=h 0; that is; tan 08 0 Next, for  908 we have sin  b=h h=h 1; that is; cos  a=h 0=h 0; that is; tan  b=a h=0 1; that is; 9 sin 908 1 > = cos 908 0 see table > ; tan 908 1
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Let us now discuss, in more detail, the statement that tan 908 1. To begin, note that, as Fig. 65 shows, as  comes CLOSER AND CLOSER to the value of 90 degrees the ratio b=a becomes GREATER AND GREATER in value. Mathematically we can say that, as  becomes vanishingly close to the limiting value of 90 degrees, the value of the ration b=a increases without bound, that is, becomes INFINITELY GREAT. The mathematical expression to indicate this situation is written as
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!908
lim tan  ! 1
72
which can be read as the tangent of  becomes in nitely great as  approaches the limiting value of 90 degrees. It should be noted that in nity is not a speci c number, but is greater than any number you name, however large. Since in nity is not a speci c number, we say that tan  is unde ned for  908. It is thus not correct to say that tan 908 equals in nity, because in nity is not a speci c value; in this case eq. (72) is really the proper statement to use. Nevertheless, it is common practice to abbreviate eq. (72) by simply writing that tan 908 1. At this point we might mention, just brie y, how the values of sin , cos , and tan , listed in the foregoing table of values, were originally found. Originally, such tables were created by drawing, as carefully as possible, right triangles for di erent values of . Then, upon measuring the lengths of the sides as accurately as possible, the true values of the ratios b=h, a=h, b=a could be approximately determined for speci c values of .