Plane trigonometry, by S.L. Loney.

A
RATIOS OF - IN TERMS OF TAN A. 121
From (2) it follows that the revolving line must lie between OR and
OS or else between OS and OP.
Both these conditions are satisfied only when the revolving line lies
A
between OR and OS, and therefore the angle - lies between
2
37r 7r
2n7r - and 2nmr -.
4 4
A
117. To express the trigonometrical ratios of A in
terms of tan A.
From equation (3) of Art. 109 we have
A
2 tan -
tan A =
1 - tan2
A 2 A.'. -ta - ta —n tan -.
2 tan A 2
A 2 A 1 1
Hence tan2 + A tan - + A 1 +
1 + tan2 A
tan2 A
A 1 \/1 +tan2A
tan + --
tan t an + tan A
A + ~/1 + tan2 - 1
2 tanA.= (1)... tan 2 = tan A............ (1).
118. The ambiguous sign in equation (1) can only
be determined when we know something of the magnitude
of A.
Ex. Given tan 15~ = 2- V/3, find tan 70~.
Putting A = 15~ we have, from equation (1), of the last article,
tan 1+ (2 - 3)2 _ 1 8 - 4^.3 - 1....
tan 2- j3 - - J-............