Plane trigonometry, by S.L. Loney.

418

TRIGONOMETRY.

We have
1-a2 2 - 2a cos
- -1+
1 - 2acos +a2 1 - 2acos 6 + a2
2 - a (ei + e-ei)
+ -1 +e-0i
-+ 2 -a (eei + e-0i)
— 1+ + —
( - aei)1 - ae-i
-1 + (1 - aei)-1 + (1 - ae-i)-1
= - 1 + 1 + aei + a2e20i +a3e3i +......
+ 1 + ae-0i + a2e-2ei + a3e-30si......
= 1 + a (eOi + e-0i) + a2 (e20i + e-20i) +......
=1 + 2a cos 0 + 2a2 cos 20 + 2a3 cos 30 +...ad inf.
The expansions of (1 - aeei)-l and (1 - ae-ei)-1 by the
Binomial Theorem are legitimate if the modulus of aeoi be
less than unity, i.e. if a be numerically < 1, but not
otherwise. (Art. 273.)
The above series is the one assumed in Art. 294.
Similarly we can deduce the series of Art. 293. For
we have
2a sin 0 1 a (ei - e-0i)
1 - 2a cos 0 + a2 i 1 - a (ei + e-ei) + a2
_ 1 a aei - ae-e6 1 1 ae
i (1 - ae0i)(l - ae-0i) i - aeei 1 - ae-0i]
- - {(1 + aei + a2e2 +...)-(1 + ae-O + a e-2ei +.)}
= 2sin+2ain 0 + 2a2 sin 30 +......ad inf.
As before this expansion is legitimate only if a < 1.