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Substituting for the first term of the second member its values (12) of 8 43, we have
2' coso x = cos 3x + cos x + 2 cos x = cos 3x + 3 cos 20.
Multiplying this equation by 2 cos x, we should have an expression for cos* x, etc. But the use of exponentials enables us not only to obtain the higher powers more expeditiously, but to find the general law of the series, which is not readily done by multiplication.
1. In the expression
1+ 2 cos x + 3 cos* x + 4 cos' 20, substitute for the powers of a their values in terms of the multiple of x, and reduce the expression to one containing simple multiples of x. Solution. From (24), 4 cos x = cos 3x
+ 3 cos a
+ 2 cos a 1
3 cos X,
Sum = cos 3x + cos 2x + 5 cos a + $
4 cos x
32 cos x 48 cos* «+ 18 coso x
proving eq. (19). 3. Prove that the expression
1 – 2a cos 0 + a' may be resolved into the two factors (1 aedi) (1 -ae-oi).
4. Resolve the expression on — 2.0" cos 0 +1 into the product of two factors, as in the last example.
Trigonometrio Forms of Imaginary Expressions.
85. It is shown in algebra that an imaginary or complex expression may be reduced to a certain number of real units plus a certain number of imaginary units. If we put
i, the imaginary unit, = V -1,
6, the number of imaginary units, the complex expression will be
(1) We have already shown (8 47) that, whatever be the numbers a and b, we can find a positive number r and an angle q, such that
rcos P = 0;
r sin p= b. If we substitute these values of r and g in (1) it will become
a+bi=r (cos o +i sin o). But equation (14) gives
cos p+ i sin prebi. Therefore a+b=repi
(2) We hence conclude: Every complex expression can be reduced to the form
reti, which is called the general form of the complex expression.
The coefficient r is called the modulus of the expression.
A yet better term, used by the Germans, is the absolute value" of the expression.
The angle p is called the argument of the expression.
- 0.9223 + 1.0962i
pe cos p=
r sin p= 1.0962, and applying the process of $47, we find
This process being purely algebraic, the angle o should be expressed in radial units. Reducing to this unit, we find
2.2703. Therefore the required general form is
- 0.9223 + 1.0962i = 1.4326e2,27034. The student who is acquainted with the geometric representation of imaginary quantities will see that the quantity r corresponds to the modulus and g to the angle of the complex expression as defined in algebra.
The geometric construction of the expression a + bi is effected by laying off the length a on the axis of X, and at the end of this length erecting a perpendicular equal to b.
If O be the origin, we shall have
86. Multiplication of complex expressions in the general form. If any two complex expressions are
rebi and qeli, we have by multiplying them
This is another complex expression of the general form of which rą is the modulus and g + 8 the argument. Hence:
The modulus of a product is equal to the product of the moduli of the factors.
The argument of a product is the sum of the arguments of the factors.
If we multiply n equal factors, each represented by redi, the result will be
= pensi. Hence :
The modulus of a power is equal to the corresponding power of the modulus of the root.
The argument of the power is the argument of the root multiplied by the index of the power.
87. Periodicity of the imaginary exponential. From the known equations (S 24)
cos (9 + 27)
= COS P,
sin (+27) = sin ,
el® + 2)i = cos (0 + 2) + i sin (ø+ 21t),
relt +20)i = reti; that is :
The value of a complex quantity remains unaltered when we increase its argument by a circumference.
Since the addition of one circumference does not change it, the addition of any number of circumferences will still leave it unchanged. Hence:
If the argument of a complex quantity increases indefinitely, the values of the quantity itself will repeat themselves with every circumference by which the argument increases.
A quantity whose value repeats itself in this way is said to be periodic.
88. Let us next inquire for what special values of g the exponential function eti will be equal to the real or imaginary unit. Considering again the equation
exi = cos o + i sin 9, we notice that sin q = 0 whenever g is a multiple of 180° or of
When the multiple of a is even, we have cos q = + 1; and when it is odd, cos Q = –1. Hence, putting
P= , 21, 31, etc., we have
etc. In order that cos o may vanish, the angle o must be 90°, 270°, 450°, etc.; that is, it must be an odd multiple of fr. Sin will
then be + 1 or -1. Putting p = 1, y = f, g = ft, etc., on both sides of the preceding equation, we have etmi +i;
e-tri = i; i; e- jäi = + i;
(6) efni = +i; e- Fri =
- i. By.squaring each of these equations we shall reproduce the alternate equations (a).
89. Roots of unity. The foregoing theory enables us to find very simple and elegant expressions for the roots of the equation
Qc" – 1= 0,
20M = 1. From the general theory of equations, the equation wo" – 1= 0, being of the nth degree, must have n roots; that is, there are n quantities which, being raised to the nth power, will produce 1.
These quantities are called the nth roots of unity.
Because 1" is always 1, whatever be n, + 1 is itself one of the nth roots of unity.
Because (-1) = 1 when n is even, - 1 is always an nth root of unity when n is even.
Hence one or two of the n roots of unity, viz. + 1 and – 1, are real; all the others are imaginary.
90. PROBLEM. To find the nth roots of unity.
Solution. Let a required root be rebi, r and being quantities to be determined. By the requirements of the problem, the nth power of this quantity must be 1. Its nth power is
(rein = prenoi = po (cos no + i sin no). In order that this expression may be equal to unity, a real quantity, the coefficient of i must vanish, and we must have
sin no = which gives
cos no = 1. Hence
go = 1, which is satisfied by supposing
n = 1. We must also have
no = 0 or 21 or 4n or 6n, etc.