Calculus 2nd Edition Rogawski

SECTION 7.9 Hyperbolic Functions 399
FIGURE 1 The St. Louis Arch has the shape
of an inverted hyperbolic cosine.
2
1
−3 −2 −1
−1
−3 −2 −1
−2
y
y = sinh x
4
3
2
y
y = cosh x
FIGURE 2 y = sinh x is an odd function,
y = cosh x is an even function.
7.9 Hyperbolic Functions
The hyperbolic functions are certain special combinations of e x and e −x that play a role
in engineering and physics (see Figure 1 for a real-life example). The hyperbolic sine and
cosine, often called “cinch” and “cosh,” are defined as follows:
sinh x = ex − e −x
, cosh x = ex + e −x
2
2
As the terminology suggests, there are similarities between the hyperbolic and trigonometric
functions. Here are some examples:
• Parity: The trigonometric functions and their hyperbolic analogs have the same
parity. Thus, sin x and sinh x are both odd, and cos x and cosh x are both even
(Figure 2):
sinh(−x) = − sinh x,
cosh(−x) = cosh x
• Identities: The basic trigonometric identity sin 2 x + cos 2 x = 1 has a hyperbolic
analog:
cosh 2 x − sinh 2 x = 1 1
The addition formulas satisfied by sin θ and cos θ also have hyperbolic analogs:
x
321
sinh(x + y) = sinh x cosh y + cosh x sinh y
x
−3
321
FIGURE 3
cosh(x + y) = cosh x cosh y + sinh x sinh y
• Hyperbola instead of the circle: The identity sin 2 t + cos 2 t = 1 tells us that
the point (cos t,sin t) lies on the unit circle x 2 + y 2 = 1. Similarly, the identity
cosh 2 t − sinh 2 t = 1 says that the point (cosh t,sinh t) lies on the hyperbola
x 2 − y 2 = 1 (Figure 3).
−2
3
2
1
−1
−2
−3
y
(cosh t, sinh t)
x
32 −1
x 2 − y 2 = 1 x 2 + y 2 = 1
y
(cos t, sin t)
• Other hyperbolic functions: The hyperbolic tangent, cotangent, secant, and cosecant
functions (see Figures 4 and 5) are defined like their trigonometric counterparts:
tanh x = sinh x
cosh x = ex − e −x
1
e x , sech x =
+ e−x cosh x = 2
e x + e −x
coth x = cosh x
sinh x = ex + e −x
1
e x , csch x =
− e−x sinh x = 2
e x − e −x
1
x