Plane trigonometry, by S.L. Loney.

476 TRIGONOMETRY.
The first of these equations states that turning a line three times in
succession through a zero angle gives the original line.
The second states that turning it three times in succession through
2w
an angle -, (i.e. altogether through 27r) gives the original line.
The third states that turning it three times in succession through an
angle -, (i.e. altogether through 47r) gives the original line.
These statements are all clearly true.
407. Multiplication of two complex quantities.
If x + iy =r (cos 0 + i sin 0),
and + iv = p (cos ~ + i sin 4),
we have
(u + iv) (x + iy) = rp [cos (0 + p) + i sin (0+ (f)].
The effect of multiplying a complex quantity x + iy
by another u + iv is therefore to turn the line representing x + iy through an angle
O) [i.e. tanand to alter its length in the ratio
1: p, i.e. 1: V/2 + v2.
Hence the multiplying of one complex quantity by
another is represented by "a turning and a stretching."