Plane trigonometry with practical applications, by Leonard E. Dickson.

Ch. VIII]

FUNCTIONS OF ANY ANGLE

105

81. Generalized notion of angle. We have been concerned
mainly with right triangles, no one of whose angles exceeds 90~.
We shall soon consider oblique triangles, no one of whose angles
exceeds 180~. But we learned in navigation to employ angles of
any size up to 360~. Likewise in surveying, where we distinguished
between the clockwise and counter-clockwise directions in which
angles may be measured, starting for example from the north
point. These remarks and especially the rotation of the telescope
of a transit in the act of measuring an angle prepare us for the
following general notion of angle:
An angle may be considered as generated by the rotation of a straight
line which first coincides with the initial side of the angle and then
rotates about the vertex of the angle until finally it coincides with the
terminal side of the angle. An angle is called positive if the rotation
is counter-clockwise, negative if the rotation is clockwise.
For example, in Figs. 59-61 are shown three angles IOT whose directions
of rotation are indicated by curved arrows. In Fig. 59, the positive angle 320~
3200
initial side initial side\ initial side
1 -40" i '400
FIG. 59 FIG. 60 FIG. 61
is generated by counter-clockwise rotation from the initial side OI to the terminal side OT. In Fig. 60, the negative angle - 40~ is generated by clockwise rotation. In Fig. 61, there has been a complete revolution clockwise
through 360~ followed by a further rotation clockwise through 40~, the combined result being the negative angle - 400~.

+3