Plane trigonometry, by S.L. Loney.

CHAPTER VI.

GENERAL EXPRESSIONS FOR ALL ANGLES HAVING A
GIVEN TRIGONOMETRICAL RATIO.
77. To construct the least positive angle whose sine is
equal to c, where a is a proper fraction.
Let OA be the initial line and let OB be drawn in the
positive direction perpendicular to OA.
Measure off along OB a distance --
ON which is equal to a units of length. /
[If a be negative the point N will lie in i
BO produced.]
Through NV draw iNP parallel to OA. With centre 0
and radius equal to the unit of length describe a circle
and let it meet NP in P.
Then AOP will be the required angle.
Draw PMI perpendicular to OA, so that
J17P ONl' a
sin A OP O = a.
The sine AOP is therefore equal to the given quantity
and AOP is therefore the angle required.