Plane trigonometry, by S.L. Loney.

[Exs. LV.] INVERSE CIRCULAR FUNCTIONS.

377

31. IfA +iB= c tan(x +iy), then
2cA
tan 2x = 2c.4
c2 _ 2 _ B2 '
32. If tan (O + 5i) = sin (x + iy), then
coth y sinh 26 = cot x sin 20.
33. If tan (a+ i/3)=i, a and 3 being real, prove that a is indeterminate and /3 is infinite.
Prove that
X5 X9
34. 1 (sinh x+sin )=x + + -.....ad inf.
X4 28
35. (cosh x + cos x) = 1 + ++.... ad inf.
# 320. Inverse Circular Functions. When a
and /3 are real and a = cos /, we defined, in Art. 237, the
inverse cosine of a to be that value of 3 which lies
between 0 and 7r, and it was pointed out that /3 was a
many-valued quantity.
If now x + yi = cos (u + vi),
then similarly u + vi is said to be an inverse cosine of
x + yi.
But since
x + yi = cos (u + vi) = cos [2n7r + (u + vi)] (Art. 311)
it follows that 2n7r + ( + vi) is also an inverse cosine of
x + yi, where n is any integer.
The inverse cosine of x + yi is hence a many-valued
function. When the many-valuedness of the inverse
cosine is considered it is written
Cos-, (x + yi).
The principal value of the inverse cosine of x + yi
is that value of 2n7r + (t + vi) which is such that either
2n7r + u or 2n7r - u lies between 0 and tr.