Page 16 - Calculus with Complex Numbers

Ayplication 1 W e can use de M oivre's theorem to obtain formulae for cos n0
sin n0 in terms of cos 0 sin 0. For example we have

= (C2 - Sl) + IiCS

where C = cos 0 S = sin 0. Hquating real and imaginal'y parts we obtain

cos 2p = (72 - sl = 2c2 1 = 1 - 1s1
-
using the identity (72 + Sl = 1 . Hence

W e also obtain similarly

sinzp = lCS = 2 cos 0 sinp.

Ayplication l W e can use the above formulae to obtain exact values for cos 45O
sin 45O as follows. lf we write 0 = 45O C = cos 45O S = sin 45O then we have

O = cos 9Oo = 2c2 - 1

from which it follows that 2C2 = 1 , andtherefore (72 = 1/2. Hence C = ulu 1/.vC,
which gives cos 45O = I/S/Y.
We also have 1 = sin 90O = ICS, which gives S = 1/2C = 1/xY, and hence
sin45O = I/vC.

I .8 E uler's form ulae for cos 0, sin 0 i n te rm s of eïkio

W e obtained the formula ei0 = cos 0 + i sin 0 in Section 1 .6. From this formula
we can derive two more formulae also attributed to Huler viz.

e -i0 e io - c -i0
io + c
cos p = , sin 0 = .
2 li
Proof Observe that

i0
e = cos p + i sin p,
e-io
= cos p - i sin p.
Now eliminate sin 0 cos 0 respectively.