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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 2r; hence, by eq. 82, there are 2r=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:
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As previously mentioned, the radian is generally used in theoretical work instead of the degree. This fact leads us now to make the following IMPORTANT NOTE: From now on, unless the degree symbol is shown, all angles will be understood to be in radians. Let us, therefore, now write eq. (81) in terms of radians instead of degrees; to do this, all we need do is substitute the total angle, 360ft degrees, into eq. (84); thus, =180 360ft 2ft total radians and therefore eq. (81) becomes, in terms of radian measurement of angles, v V sin 2ft 86
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in which the angular velocity of the wave (see discussion following eq. (81)) is now 2f radians per second. We now have one nal change in notation to make, as follows. It is universal practice to represent the quantity 2f by the small Greek letter omega, written ! ; thus it will always be understood that ! 2f angular velocity of radius vector in radians per second and thus, t seconds after starting at t 0, the radius vector V will have covered a total angular distance of !t radians, and thus eq. (86) now takes the standard form v V sin !t 87
where v and V have the meanings de ned in connection with eq. (81), and where ! 2f , where f is the frequency in cycles per second (Hz). A corresponding sine wave of current will of course have the same basic form i I sin !t 88
where now i is the instantaneous current and I is the maximum (peak) value of current. For the next part of our discussion, let us begin by returning to the table of values listed in section 5.3. Note that the table covers the range of angular values from  08 to  3608. What we now wish to do is to write the same table (omitting a few values here and there) in terms of radians instead of degrees. This can be done as follows. First, making use of eq. (84), you can verify that 08 0 rad: 308 =6 rad: 458 =4 rad: 608 =3 rad: 908 =2 rad: 1208 2=3 rad: 2108 7=6 rad: 1358 3=4 rad: 2258 5=4 rad: 1508 5=6 rad: 2408 4=3 rad: 1808  rad: 2708 3=2 rad: 3008 5=3 rad: 3158 7=4 rad: 3308 11=6 rad: 3608 2 rad:
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