Plane trigonometry, by S.L. Loney.

394

TRIGONOMETRY.

[Exs, LVIII.]

Raise all these quantities to the power / -1; thus
e- 2r=e-~ =e- =........ 27r=47r=6r=.......
16. For all values of ' we have
cos (0 - r) + i sin (0 - 7r) cos (0 + r) + i sin (0 + 7r),
so that ei (-) = ei (- r).
Hence 0- r=0+7r, i.e. 7r=0.
17. If 0 and 0 be the principal values of the amplitudes of two
complex numbers x and y, prove that
log xy = log x + log y + 2n7ri,
where n is -1, 0, or +1 according as 0+0 is >7r, greater than -r and
not greater than 7r, and not greater than - 7r, respectively.