Plane trigonometry, by S.L. Loney.

4

TRIGONOMETRY.

For any position between OB and OA', such as OP2,
the angle AOP2 through which it has turned is greater
than a right angle.
For any position OP3, between OA' and OB', the
angle traced out is AOP3, i.e. AOB + BOA' + A'OP3, i.e.
2 right angles + A'OP, so that the angle described is
greater than two right angles.
For any position OP4, between OB' and OA, the angle
turned through is similarly greater than three right angles.
When the revolvinr line has made a complete revolution, so that it coincides once more with OA, the angle
through which it has turned is 4 right angles.
If the line OP still continue to revolve, the angle
through which it has turned, when it is for the second
time in the position OP1, is not AOP1 but 4 right angles
+AOP1.
Similarly when the revolving line, having made two
complete revolutions, is once more in the position OP2,
the angle it has traced out is 8 right angles + A OP2.
7. If the revolving line OP be between OA and OB
it is said to be in the first quadrant; if it be between OB
and OA' it is in the second quadrant; if between OA' and
OB' it is in the third quadrant; if it is between OB' and
OA it is in the fourth quadrant.
8. Ex. What is the position of the revolving line when it has turned
through (1) 225~, (2) 480~, and (3) 1050~?
(1) Since 225~=180~+45~, the revolving line has turned through
45~ more than two right angles and is therefore halfway between OA'
and OB'.
(2) Since 480 =360~+120~, the revolving line has turned through
120~ more than one complete revolution, and is therefore between OB
and OA', and makes an angle of 30~ with OB.