788. Cosine. The ratio of the distance of M from 0 (or of PNOY) to OP is the cosine of A, written cos A, i. e.,

The cos A is positive by definition when M lies to the right, and negative when M lies to the left, of the axis of pure imaginary, YOY'.

789. The expression of a + ib in terms of the modulus r and argument A is (cosA+i sinA).

Hence, a + b = r cos A + ir sind = r (cosA+i sinA.) If r

the point P will move about the

is fixed and A varies from 0 to 2
circumference of the circle PBAP counter clock-wise.

In the first case, a = 0, b = 1, and P is at B; in the second case,

a = — 1, b

--

=

0, and P is at A'.

The sign of in (1) is always taken positive in the expression a+ibr (cosA+ i sin A), and is called the absolute value of a +ib and represents the distance of the corresponding point P from 0. Of two complex numbers, that is the greater whose corresponding point is at the greater distance from 0.

The expression cos A+ i sin A has the same kind of geometrical meaning as and, which are simply particular cases of it, namely, in the first case A = 0, cos 0 + i sin 0 = +1, and the second case A =π, COST+ i sin = - - 1.

790. Geometrical Representation of the n Roots of 1 (Fig. 2).

1. Two roots of 1, x = v 1.

Here cos A+ i sin A 1, hence, A = 0 or and r = 1.

The roots1 and 1 are represented by the points P, and P which bisect the circumference.

3

A =

The points P, Q, Q, divide the circumference into three equal parts. 3. The biquadratic roots of unity are (8782)

which correspond respectively to the points P, B, P, B', which divide the circumference into four equal parts.

4. Then roots of 1 are given in the table in 779. It will be noticed on comparing the arguments for the various roots that the difference between any two consecutive arguments is. Therefore the n points which correspond to the n roots will divide the circumference into n equal parts.

4. Give a geometrical representation of the quinary, sextenary,

and septenary roots of unity.

CHAPTER V

THEORY OF EQUATIONS

PROPERTIES OF EQUATIONS

791. First Property.-If r is a root of f(x) = 0, then f(x) is exactly divisible by x r (8764).

792. Second Property.-If the first member of the equation f(x) = 0 is divisible by x r, r is a root of f(x) = 0 (2765).

793. Third Property.-Every equation of the nth degree has n roots and no more. Consider the equation

If a,, real or imaginary, is a root of f(r) = 0, then f(x) is exactly divisible by x- - ar Let the quotient be Q, (c), then one may write

1

f(x) = (x — a ̧) Q2 (x).

If a, is another root, real or imaginary, of f(x) = 0, it must be a root of Q(x) = 0, hence Q,(x) is exactly divisible by x-a, and therefore we may write

f(x)=(xa) (x — α) Q2 (x)

2

where the second quotient Q, (c) is of the degree n—2. Continuing this process we obtain

f(x) = (x — a ̧) (x − a) . . . . (x — a„).

Since ƒ(x) vanishes if x is made successively equal to the n values , ༦ , . an, f(x) has n roots; and the equation has no more than n roots, for, if any value of x, which is not one of the n

which can not vanish since none of the factors of the product vanishes.

Q,(r), in

794. Fourth Property. To depress an equation. If one of the roots of the equation f(r)=0 is known, the equation may by division be depressed to an equation of the next lower degree, which contains the remaining roots. Thus, 793, contains all of the n roots of f(x) = 0, except a.. the degree of the depressed equation is (n—r). two are known the depressed equation is a quadratic, which can be readily solved.

Thus, if

roots are known, If all the roots but

795. Fifth Property. - To form an equation whose roots are

it is evident that an equation can be formed by subtracting each root from ≈ and placing the continued product of the binomial factors equal to 0.

796. Sixth Property.-The relations between the coefficients of f(x) and the roots of f(x) = 0. Let the equation be

ƒ(x)=x"+p1xn−1+P ̧‚xn−2+ . + Pn = 0,

Equating the coefficients of like powers of x (2666),

and S is the sum of the products of the n roots taken two at a time

2

Thus. The sum of the roots equals the coefficient of the second term with the sign changed.

The sum of the products of the roots taken two at a time is the coefficient of the third term.

The sum of the products of the roots taken three at a time is the coefficient of the fourth term with its sign changed, and so on.

The product of the roots equals the last term taken with the positive or negative sign according as the number of roots is even or odd.

797. Although the relations just derived will not in general determine the roots of any proposed equation, yet by means of them we can derive many relations which are of value in solving various problems.

Let 1, 2, 3, be the roots of the equation

Hence, if in a cubic p-2p, is negative, the roots of the equation can not all be real.

798. The relations connecting the roots and coefficients of an equation sometimes enable one to find all the solutions of the equations in case the roots are required to satisfy assigned relations.

799. The Sixth Property.-In an equation with real coefficients, imaginary roots enter in pairs.

Let the coefficients in the equation

be real; then, if a+bi is a root of f(x) = = 0, we are to prove that bi is also a root.

a

Since abi is by hypothesis a root of f(c) = 0, then on replacing a by a + bi, this equation will take the form P+Qbi = 0, where P and Q involve even powers of b. Because, for example, if " is expanded, where x = a + bi, the even powers of bi give rise