Plane trigonometry, by S.L. Loney.

378

TRIGONOMETRY.

This principal value is denoted by cos-l (x + yi).
We have then
Cos-l (x + yi) = 2nTr + cos-l (x + yi).
321. Similarly if
x + yi = sin (u + vi) = sin [{nr (- 1)" (u + vi)},
then n7r + (- 1)" (u + vi) is an inverse sine of x + yi. It is
a many-valued quantity and is denoted by Sin-l (x + yi).
Its principal value is such that its real part lies between
7 T
- 7 and, and is denoted by Sin-l (x + yi).
2 2'
We then have
Sin-l (x + yi) = nwr + (- 1)n sin-' (x + yi).
Similarly tan-l (x + yi) and Tan-l (x + yi) are defined,
so that the principal value of Tan-l (x + yi) is such that
its real part lies between - and +, and
2 2
Tan-l (x + yi) = nor + tan-l (x + yi).
Similarly
Sec-l (x + yi)= 2n7r + sec-l (x + yi),
Cosec-l (x + yi) = nor + (- 1)7 cosec-1 (x + yi),
and Cot-l (x + yi) = n7r + cot-l (x + yi).
* 322. We shall henceforward use sin-l, Sin-1,
cos-l, Cos-l,... with the meanings above assigned.
* * 323. Inverse hyperbolic functions. If x=coshy
then similarly, as in Art. 320, we write y = cosh-1 x.