Plane trigonometry, by S.L. Loney.

370

TRIGONOMETRY.

Similarly the quantity
eY + e-y
2
is called the hyperbolic cosine of y and is written
cosh y.
[It will be observed that the values of sinh y and cosh y are obtained
from the exponential expressions for sin y and cos y by simply omitting
the i's.]
The hyperbolic tangent, secant, cosecant, and cotangent
are obtained from the hyperbolic sine and cosine just as
the ordinary tangent, secant, cosecant, and cotangent are
obtained from the ordinary sine and cosine.
sinh y ey - e-y
Thus tanh y =cosh y
1 2
^cosech sinh y ey - e-
1 2
sech y = -, = --
sech cosh y ey e-y'
I ey + e-Y
and coth y = nh = - ev tanh y ey - e-y
The hyperbolic cosine and sine have the same relation
to the curve called the rectangular hyperbola that the
ordinary circular cosine and sine have to the circle.
Hence the use of the word hyperbolic.
314. From Arts. 312 and 313 we clearly have
cos (yi) = cosh y,
and sin (yi) = i sinh y.

So

tan (yi) = i tanh y.