Plane trigonometry, by S.L. Loney.

DE MOIVRE'S THEOREM.

315

Ex. 1. Simplify
(cos 30 + i sin 30)5 (cos 0 - i sin 0)3
(cos 50 + i sin 50)7 (cos 20 - i sin 20)5
We have cos 30 + i sin 3 =(cos 0 + i sin 0)3,
cos - i sin 0=cos (- ) +i sin (- 0)=(cos 0 +i sin O)-1,
cos 50 + i sin 50 = (cos 0 + i sin 0)5,
and cos 20 - i sin 20 = cos (- 20) + i sin ( - 20)= (cos 0 + i sin )-2.
The given expression therefore
(cos + i sin 0)15 (cos 0 + i sin 0)-3
(cos 6 + i sin 0)35 (cos 0 + i sin 0)-10
= (cos 0 + i sin 0)-13= cos 130 - i sin 130.
1 1
Ex.2. If 2 cos 0=x+- and 2 cos =y +-,
x y
prove that 2 cos (mno + no) = xy + -x -
We have x2 - 2x cos 0 = - 1.. ( - cos 0)2= -1 + cos2 0= - sin2 6... = cos 0+isin0,
so that xnm = cos mO + i sin nwO,
and - = cos m - i sin I0.
Similarly y = cos q + i sin q,
so that yJn = cos n + i sin n0,
and - = cos nf - i sin nq.
ynl. xmyn + xMy
= (cos mO + i sin mO) (cos n? + i sin np)
+ (cos mO - i sin mO) (cos nq - i sin np)
= cos (me + no) + i sin (mo + nq)
+ cos (mO + no) - i sin (mO + no)
= 2 cos (mO + n).
Similarly it could be shewn that
xm ynj
- +X 2 cos (mn - n7).
YA X