Plane trigonometry, by S.L. Loney.

CHAPTER XVIII.

INVERSE CIRCULAR FUNCTIONS.
237. IF sin 0=a, where a is a known quantity, we
know from Art. 82, that 0 is not definitely known. We
only know that 0 is some one of a definite series of
angles.
The symbol "sin-l a" is used to denote the smallest
angle, whether positive or negative, that has a for its sine.
The symbol " sin-' a" is read in words as " sine minus
one a," and must be carefully distinguished from 1
sin a
which would be written, if so desired, in the form (sin a)-l.
It will therefore be carefully noted that "sin-l a" is an
angle and denotes the smallest numerical angle whose
sine is a.
So " cos-l a" means the smallest numerical angle
whose cosine is a. Similarly "tan-l a," "cot-l a," "cosec-1 a," "sec- a," " vers-~ a," and "covers-l a," are defined.
Hence sin-l a and tan-l a (and therefore cosec-l a and
cot-l a) always lie between - 90~ and + 90~.
But cos-l a (and therefore sec-l a) always lies between
0~ and 180~.