Plane trigonometry, by S.L. Loney.

380

TRIGONOMETRY.

Squaring and adding, we have
1 = sin2 x cosh2 y + cos2 x sinh2 y = sin2 x (1 + sinh2 y) + cos2 x sinh2 y
= sin2 x + sinh2 y,.sinh2 y = cos2 x.
Hence from (2) we have cos2 x= sin 0, assuming sin 0 to be positive.
Therefore, since x is to lie between - and + (Art. 321),
2 2
we have cos x = + ^sin 0, and hence x= cos-1 (~sin 0).
The equation (2) then gives
sinh y = + / sin 0,
so that e2y - 2ey Jsin 8 = 1.
i.e. ey = sin 8+1 + sin 0,
i.e. y =log [Vsin 0 + / + sin 0].
Ex. 2. Separate into its real and imaginary parts the quantity
tan-' (a + pi).
Let tan-l (a + pi)= (x + yi), so that tan (x + yi) = a + pi,
and tan (x - yi) = a - pi.. tan 2x =tan (x + yi) +(x- yi)}
(a+gi)+(a-/3i) 2a
1 - (a+ pi) (a -i)- 1-a2-p2
2a
x=~tan-l' a2 --2 ti -1 - 2 2 '
Again tan (2yi) = tan [(x + yi) - (x - yi)]
(a + 0i) - (a - ) _ 2/3i
1 +(a + i) (a-pi) l+a2 + P2
e2y 1l+a2+-2+20_ (1 +)2+a2
e-2y l+a2+2- 2~ - (1 -i)2+a2
y=jlog +