James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

260 Chapter 3 Differentiation Rules
d
FIGURE 5
Idealized ocean wave
L
Another application of hyperbolic functions occurs in the description of ocean waves:
The velocity of a water wave with length L moving across a body of water with depth d
is modeled by the function
v −Î tL
2 tanh S 2d
L
D
where t is the acceleration due to gravity. (See Figure 5 and Exercise 49.)
The hyperbolic functions satisfy a number of identities that are similar to well-known
trigonometric identities. We list some of them here and leave most of the proofs to the
exercises.
Hyperbolic Identities
sinhs2xd − 2sinh x
cosh 2 x 2 sinh 2 x − 1
coshs2xd − cosh x
1 2 tanh 2 x − sech 2 x
sinhsx 1 yd − sinh x cosh y 1 cosh x sinh y
coshsx 1 yd − cosh x cosh y 1 sinh x sinh y
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The Gateway Arch in St. Louis was
designed using a hyperbolic cosine
function (see Exercise 48).
ExamplE 1 Prove (a) cosh 2 x 2 sinh 2 x − 1 and (b) 1 2 tanh 2 x − sech 2 x.
SOLUTION
(a)
2
D 2S
2
cosh 2 x 2 sinh 2 x −S e x 1 e 2x
− e 2x 1 2 1 e 22x
4
e x 2 e 2x 2
D
2
2 e 2x 2 2 1 e 22x
4
− 4 4 − 1
(b) We start with the identity proved in part (a):
cosh 2 x 2 sinh 2 x − 1
If we divide both sides by cosh 2 x, we get
y
P(cos t, sin t)
1 2 sinh2 x
cosh 2 x − 1
cosh 2 x
or 1 2 tanh 2 x − sech 2 x ■
FIGURE 6
O
Q x
≈+¥=1
The identity proved in Example 1(a) gives a clue to the reason for the name “hyperbolic”
functions:
If t is any real number, then the point Pscos t, sin td lies on the unit circle x 2 1 y 2 − 1
because cos 2 t 1 sin 2 t − 1. In fact, t can be interpreted as the radian measure of /POQ
in Figure 6. For this reason the trigonometric functions are sometimes called circular
functions.
y
P(cosh t, sinh t)
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