Plane trigonometry, by S.L. Loney.

322

TRIGONOMETRY.

[Exs. XLVIII.]

17. Prove that 4/a + bi a - bi
has n real values and find those of
+ n r s ss /- 3.
18. Prove that the n nth roots of unity form a series in G. P.
19. Find the seven 7th roots of unity and prove that the sum of their
nth powers always vanishes unless n be a multiple of 7, n being an integer,
and that then the sum is 7.
273. Binomial Theorem for Complex Quantities.
It is known that for any real values of n and z, provided
that z be less than unity, we have
(1 +\ )n = l + n(n-l) z + (n- 1)(n-2) z3 +
(1+^=1+^4- \ +z )+ 1^ + - + 1**.1. 2 1.2.3......... (1).
When z is complex (= x + y - 1) and n is a positive
integer, the ordinary proof applies and the theorem (1) is
still true.
When z is complex, and n is a fraction or negative, it
can be shewn that
1 +nz + - z......... (2)
1. 2
is one of the values of (1 + z)n, provided that the modulus
of z, i.e. x2 + y2, is less than unity. When this modulus
is equal to unity, the theorem is only true (1) when n is
positive, and (2) when n is a negative fraction and z is not
equal to - 1.
The proof is difficult and beyond the range of the
present book. We shall therefore assume the result.
The student may hereafter refer to Hobson's Trigonometry, Arts. 211 and 212.