Plane trigonometry, by S.L. Loney.

LOGARITHMS OF COMPLEX QUANTITIES. 387
334. Ex. 1. Resolve into its real and imaginary parts the expression
Log sin (x+ yi).
Let Log sin (x + yi)= u +vi, so that
et+i = sin (x + yi) = sin x cos yi + cos x sin yi
ey + e-y ey - e —
= sn x 2 +i cos x 2....................(1).
As in Art. 267 let the right-hand side of this expression equal
r [cos (2nwr + 0) + i sin (2n7r + 0)],
so that
/~ * /ev + e-y\2 S eV - e-Y\ 2
r=+ sin2 x ) + cos2 )
2 2
= V ~(eTY + e-~2) - 2 cos 2x
i /o —io —o —o" / cosh 2y/ - cos 2x
= 1 /2 cosh 2y - 2 cos 2x = cosh2y -cos2x
r~ gey - e-y1
and 0 = tan-1 cot x ey + Y= tan e [cot x tanh y],
with the usual restriction of Art. 267.
We have then from (1)
eu (cos v + i sin v) = r [cos (2n7r + ) + i sin (2n7r + 0)].
Hence eu = r, so that = loge r,
and v = 2nr +..'. Log sin (x+yi)=u+vi=log r +(2n7r+ ) i
= loge [ - co 2 + i [2n7r + tan-1 (cot x tanh y)].
By putting n equal to zero, we have the principal value of
Log sin (x + iy).
Ex. 2. Find the general value of Log (- 3).
Let x + yi = Log (-3), so that
ex+yi = 3
Put - 3 =r {cos (2n7r + 0) + i sin (2nr + )},
as in Art. 267.
Then we have r = 3 and = 7r.
25-2