Plane trigonometry, by S.L. Loney.

474

TRIGONOMETRY.

Hence the length of OP represents the modulus and
MOP the principal value of the Amplitude of x + iy.
(Art. 265.)
404. Addition of two complex quantities.
Let OP represent the complex quantity x + iy and
OQ represent u + iv, so that y
ON=x, NP = y, OM = u,
and MQ = v. Q..-R
Complete the parallelogram — Ss
OPRQ, and draw RL perpendicu- o M N L X
lar to OX and PS perpendicular to RL.
Since PR is equal and parallel to OQ, we have
NL = PS = OM, and SR = MQ.
Hence OL = ON + NL = x + u,
and LR= LS + R =y +v.
Therefore OR represents the complex quantity
x + u+ i (y +v),
so that the sum of two complex quantities is represented by the diagonal of the parallelogram whose two
adjacent sides represent the two given complex quantities.
405. Let
x + iy = r (cos 0 + i sin 0),
as in Art. 265.
Then
(cos a + i sin a) (x + iy) = r (cos a + i sin a) (cos 0 + i sin 0)
r [cos (a + 0) + i sin (a + 0)].........(1).
Now r [cos 0 + i sin 0]
means, with our interpretation, a line of length r drawn at,
an angle 0 with OX.