082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.4 Review of some common engineering functions and techniques 101
Solution (a) cosh3 = 10.07
(b) sinh(−2) = −3.627
(c) tanh(1.6) = 0.9217
(d) sech(−2.5) = 1/cosh(−2.5) = 0.1631
(e) coth1 = 1/tanh1 = 1.313
(f) cosech(−1) = 1/sinh(−1) = −0.8509
Graphs of the functions sinhx, coshx and tanhx can be obtained using a graphics
calculator.
Hyperbolicidentities
Severalidentitiesinvolvinghyperbolicfunctionsexist.Theycanbeverifiedalgebraically
usingthe definitions given, and are listedfor reference.
cosh 2 x −sinh 2 x = 1
1 −tanh 2 x = sech 2 x
coth 2 x −1 = cosech 2 x
sinh(x ±y) = sinhxcoshy ±coshxsinhy
cosh(x ±y) = coshxcoshy ±sinhxsinhy
sinh2x = 2sinhxcoshx
cosh2x = cosh 2 x +sinh 2 x
cosh 2 x =
sinh 2 x =
cosh2x +1
2
cosh2x −1
2
Note also that
e x = coshx +sinhx
e −x = coshx −sinhx
Hence any combination of exponential terms may be expressed as a combination of
coshx and sinhx, and viceversa.
Example2.20 Express
(a) 3e x −2e −x intermsofcoshx and sinhx,
(b) 2sinhx +coshx intermsofe x and e −x .
Solution (a) 3e x −2e −x = 3(coshx +sinhx) −2(coshx −sinhx) = coshx +5sinhx.
(b) 2sinhx+coshx=e x −e −x + ex +e −x
= 3ex −e −x
.
2 2