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CHAPTER 5 Sinusoidal Waves. rms Value
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So far in our work we ve always measured angles in units of degrees, a full circle being divided into 360 equal degrees, as you know. There is, however, another unit of angular measurement called the radian that we should be familiar with. The radian is simply a unit of angular measurement, like the degree, but use of radians, instead of degrees, has certain advantages in theoretical work. The radian unit of angular measurement is de ned in references to a circle, just as the degree is. The radian will now be de ned with the aid of Fig. 84.
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Fig. 84
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In Fig. 84, let O denote the center of any circle of radius r, let a denote the length of the arc of the circle cut o by the two radii, and let  denote the ANGLE subtended by the arc a, as shown. Then the angle in RADIANS is de ned to be equal to the ratio of the arc length to the radius, that is,  angle in radians arc length a radius r
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The ratio a=r does not, of course, depend upon the size of a particular circle we might draw, but only upon the value of the angle ; that is, a is always directly proportional to r. Thus, using the standard notation of Fig. 84, we have the basic de nition that the angle , in radians, is equal to  a r 82
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Figure 84 and eq. (82) should be committed to memory. Next, the relationship between RADIANS and DEGREES can be found by considering a FULL CIRCLE, as follows. In a full circle we have a circumference of circle 2r; hence, by eq. 82, there are 2r=r 2 radians in a full circle. Since there are also 360 degrees in a full circle, we have that the ratio of radians to degrees is 2 to 360 or  to 180, thus giving us the important relationship radians  degrees 180 83
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which should also be committed to memory. From eq. 83 we thus have the following two conversion formulas    radians degrees 84 180   180 degrees radians 85  Note that for 1 radian eq. (85) gives the value, degrees 180= 1 57.295 78, that is, one radian equals 57.295 78 degrees 57817 0 45 00 . Thus the radian is a much larger unit of
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CHAPTER 5 Sinusoidal Waves. rms Value
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angular measurement than the degree. In any case, the conversion from one to the other is easily done with the aid of eqs. (84) and (85) and a calculator: for example, to convert 240 degrees to radians we use eq. (84); thus, radians =180 240 4:188 79, answer: