Plane trigonometry, by S.L. Loney.

272

TRIGONOMETRY.

238. The quantities sin-l a, cos-1 a, tan-l a,... are
called Inverse Circular Functions.
The symbol sin- a is often, especially in foreign
mathematical books, written as "arc sin a"; similarly
cos-l a is written " arc cos a," and so for the other inverse
ratios.
239. When a is positive, sin-l a clearly lies between
0~ and 90~; when a is negative it lies between - 90~ and
00.
Ex. sin-2 =30~; sin-1 =- 600.
2 2
When a is positive, there are two angles, one lying
between 0~ and 90~ and the other lying between -90~
and 0~, each of which has its cosine equal to a. [For
example both 30~ and -30~ have their cosine equal to
/3.] In this case we take the smallest positive angle.
Hence cos-l a, when a is positive, lies between 0~ and 90~.
So cos-1 a, when a is negative, lies between 90~ and
180~.
Ex. cos- = 45~; cos-1 - =120~.
When a is positive, the angle tan- a lies between 0~
and 90~; when a is negative, it lies between - 90~ and 0~.
Ex. tan-1/3 = 60~; tan-1 (-1)= -45~.
3 12 16
240. Ex. 1. Prove that sin- - cos-1 -= sin-1 6
5 l3 b5
3 3
Let sin-l= a, so that sin a=
and therefore cos a= - 5
9_ 4
and therefore COS a= 1 -25 =.
1r ^0 os=