Plane trigonometry, by S.L. Loney.

COMPLEX QUANTITIES. CIRCULAR FUNCTIONS. 367

Hence for all values of x, real or complex, we have
eexi + e- i exi e-i
cos x 2 and sinx=-
These results are known as Euler's Exponential Values.
309. We can now shew that the Addition and
Subtraction Theorems hold for imaginary angles, i.e. that,
whether x be real or complex, then
sin (x + y) = sin x cos y + cos x sin y,
cos (x + y) = cos x cos y - sin x sin y,
sin (x - y) = sin x cos y - cos x sin y,
and cos (x - y) = cos x cos y + sin x sin y.
Since
exi + e-xi exi - e~i
cos x = and sin x =
2 2i
we have sin x cos y + cos x sin y
exi -e-xi eyi + ei + e-xi ei - e-yi
2i 2 2 2i
exi. 2ei - e-xi. 2e-Yi e(x+Y)i _ e-(x+y
= ---.4i - = 2i-.- (Art. 305)
4i 2i
= sin (x + y).
Similarly the other results may be proved.
310. It follows that all formulae which have been
proved for real angles and which are founded on the
Addition and Subtraction Theorems are also true when
we substitute for the real angle any complex quantity.
For example, since
cos 30 = 4 cos3 0 - 3 cos 0,