Plane trigonometry, by S.L. Loney.

388 TRIGONOMETRY.
Hence 3 {cos (2nr + 7r) +i sin (2n7r + r)}
= ex+yi= ez. eyi =ex {cos y + i sin y}.
Hence ex=3, so that x= loge 3, and y = 2n7r +r..'. Log (- 3) =loge 3 + (2nr + r) i.
The principal value, obtained by putting n equal to zero, is
loge 3 + 7ri.

EXAMPLES. LVII.
Prove that
1. log(cos + i sin 0)-iO, if - r < 7r. 2. log(-l)=7riL
3. log(-i)= -i.
4. log (1 + cos 20 +i sin 28) =log, (2 cos 0) + i, if - 7r < 7r.
5. log tan (+ i) =itan sin hx.
-, /., 1, /cosh 2y+ cos 2x\),, -,
6. log cos (x + yi) = 2log (cosh 2 cos 2) + i tan-' (tan x tanhy).
sin (x +yi)
7 log sin +i) = 2i tan- (cot x tanh y).
cos (x- yi)
8. log cos ( yi) = 2i tan-' (tan x tanh y).
cos(x+ yi)
9. ilog — = 7 - 2 tan-l x.
10. log (1 + i tan a) = loge sec a + ai, where a is a positive acute angle.
1 1.0 rI11. log (1- )=loge (cosec2) (+ - 2)
a+ bi b
12. log +bi = 2i tan -.
a- bi a
13. Log (-5)= loge 5 + (2n + 7r) i.
14. Log (1+i) = log, 2+i (2nr+).
15. Find the value of log log sin (x +yi).