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CHAPTER 6 Algebra of Complex Numbers
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which is called De Moivre s Theorem ( dah MWAH vrah ).* The theorem is true for all positive and negative, integral and fractional, values of n. Problem 99 Two complex numbers are EQUAL only if their REAL PARTS are equal and their IMAGINARY PARTS are equal (see problem 88). Using this fact and De Moivre s theorem, nd (a) a trigonometric identity for cos 2x, (b) a trigonometric identity for sin 2x. Problem 100 Find the value of 2 cos 488 j sin 488 raised to the fth power. Problem 101 Find the value of 1 4 cos 178 j sin 178 3 :
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Problem 102 Making use of eq. (161), and using the same procedure as in problem 99, nd (a) the trigonometric identity for cos x y , (b) the trigonometric identity for sin x y .
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a jb A cos  j sin  167
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A complex number can be written in the rectangular form using the notation of either eq. (156) or eq. (158); thus in which the MAGNITUDE A and ANGLE  are given by eqs. (143) and (144) in section 6.4. Now let us raise both sides of the last equation to a power n; thus a jb n An cos  j sin  n which, by virtue of eq. (166), can also be written in the form a jb n An cos n j sin n 168
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As in the case of eq. (166), eq. (168) is valid for all positive and negative, integral and fractional, values of n. Let us, however, rst consider the case where n is any positive or negative INTEGER (whole number). The following two problems will illustrate the procedure for the case where n is a positive or negative integer. Problem 103 Using eq. (168), show that 3 j2 7 4449:06 j6553:97, approximately.
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* Named for the French mathematician Abraham De Moivre (1667 1754).
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CHAPTER 6 Algebra of Complex Numbers
Problem 104 Using eq. (168), show that 6000 3 j4 5 0:1457 j1:9144; approximately:
Now (in preparation for nding the root of a complex number) let us be reminded that the sine and cosine are PERIODIC functions having a period of 360 degrees (2 radians). This is expressed (in degrees) by eqs. (79) and (80), in section 5.3; thus sin 8 sin  360k 8 cos 8 cos  360k 8 169 170
where k 0; 1; 2; 3; . . . ; that is, where k is any positive INTEGER. If  is measured in radians the corresponding equations are sin  sin  2k cos  cos  2k 171 172
Since k is an INTEGER* the above equations are true for all values of k; this is because if  is increased or decreased by any INTEGRAL MULTIPLE of 3608 (or 2 radians), the angle simply returns to its original position, as illustrated in Fig. 111.
Fig. 111
Thus, since k is an INTEGER, cos 8 cos  360k 8 and sin 8 sin  360k 8 and hence eq. (167) can be written in the form a jb A cos  360k 8 j sin  360k 8 175 174 173 {
Now let n be any given positive INTEGER, and let us, using De Moivre s theorem, raise both sides of the last equation to the 1=n power; thus,    !  k 8  k 8 j sin 360 176 a jb 1=n A1=n cos 360 n n n n
* k is a positive INTEGER, k 0; 1; 2; 3; . . . ; in all our work in this section. { From inspection of Fig. 111, note that  360k 8  360k 8, because k is an integer.