Plane trigonometry, by S.L. Loney.

310

TRIGONOMETRY.

It is customary to take the positive square root of
x2 + y2 and hence r is known.
From (1) and (2) we then have
cos 0 = and sin =
1X2 + y2 x2 + y2
Whatever be the values of x and y, there is one value
of 0, and only one value, lying between - r radians and
+ tr radians which satisfies these two equations.
The quantity x + y V -l can therefore always be
expressed in the form r (cos 0 + J - 1 sin 0).
Def. The quantity + Jx + y2 is called the Modulus
of the complex quantity, and that value of 0 (lying
between - r and + 7r) which satisfies the relations
x y
cos 0 = and sin 8 =
+ 2/x2 + y2 + V/2 + y2
is called the principal value of the Amplitude of
x+y -l.
266. Ex. 1. Express in the above form the quantity 1 + -/1l.
Here 1 + -l = r (cos o +,/ - 1 sin ),
so that r cos 0=1,
and r sin = 1.
We therefore have r= + /1 += + ^/2,
1 1
and then cos 0 = - and sin 0 =,
so that O=.
4
Hence 1 + /- =^/2 [cos - + V-1i sin],
so that ^/2 is the modulus and 4 is the principal value of the amplitude
of the given expression.