DEFINITION
For any x real or complex the expressions
(ex - e−x)/2 and (ex + e−x)/2
are respectively defined as hyperbolic sine x and hyperbolic cosine x and are written as sinh x and cosh x respectively.
That is, sinh x = (1/2)(ex – e–x),cosh x = (1/2)(ex + e–x).
Similar to the trigonometric functions, we define the remaining four hyperbolic functions as follows:
| tanh x = | sinh x
cosh x = ex - e−x ex + e−x |
| sech x = | 1
cosh x = 2 ex + e−x |
| coth x = | cosh x |
We now extend this definition to complex values of the variable. We agree that the above equations define cosh x and sinh x for all values of x real or complex.
That is,
(i) cos ix = cosh x
(ii) sin ix = i sinh x
(iii) cosh ix = cos x
(iv) sinh ix = i sin x
We have from Euler’s exponential formula
eix = cos x + i sin x and e-ix = cos x – i sin x
By adding and subtracting these two, we obtain
cos x = eix + e-ix
2 and sin x = eix – e-ix
2i
Formulae for the Sum and Difference
- sinh(x ± y) = sinh x cosh y ± cosh x sinh y
- cosh(x ± y) = cosh x cosh y ± sinh x sinh y
- tanh(x ± y) = (tanh x ± tanh y) / (1 ± tanh x tanh y)
Formulae for Multiple Angles
- sinh 2x = 2 sinh x cosh x
- cosh 2x = 2 cosh²x − 1
or 1 + cosh 2x = 2 cosh²x - cosh 2x = 1 + 2 sinh²x
or 1 − cosh 2x = −2 sinh²x - tanh 2x = (2 tanh x) / (1 + tanh²x)
- sinh 2x = (2 tanh x) / (1 − tanh²x)
- cosh 2x = (1 + tanh2x) / (1 - tanh2x)
- sinh 3x = 3 sinhx + 4 sinh3x
- cosh 3x = 4 cosh3x – 3 cosh x
- tanh 3x = (3 tanh x + tanh3x) / (1 + 3 tanh2x)
Some More Formulae
- sinh2x – sinh2y = sinh(x + y) sinh(x – y)
- cosh2x + sinh2y = cosh(x + y) cosh(x – y)
- cosh2x – cosh2y = sinh(x + y) sinh(x – y)
- (cosh x + sinh x)n = enx = cosh nx + sinh nx
- (cosh x – sinh x)n = e–nx = cosh nx – sinh nx
- cos2h x – sin2h x = 1
- cos2h x + sin2h x = cosh 2x
- 1 – tanh2 x = sech2 x
- 1 – coth2 x = – cosech2x
EXPANSION OF sinh x AND cosh x
sinh x = (1/2) (ex − e−x) = x + x3/3! + x5/5! + x7/7! + …
cosh x = (1/2) (ex + e−x) = x2/2! + x4/4! + x6/6! + …
These expansions easily follow from the corresponding expansions of sin x and cos x on replacing x by ix.
INVERSE HYPERBOLIC FUNCTIONS
If sinh y = x, then y is said to be in the inverse hyperbolic sine of x and it is written as sinh−1x = y.
Similarly, we can define cosh−1x, tanh−1x etc.
Example: The value of sinh−1x is equal to
(A) log(x − √x2 + 1)
(B) log(x + √x2 + 1)
(C) log(x + √x2 − 1)
(D) none of these
Solution: (B).
Let sinh−1x = y. Then
x = sinh y = (ey − e−y) / 2
⇒ 2x = ey − e−y
⇒ e2y − 2x ey − 1 = 0
⇒ ey = [2x ± √4x2 + 4] / 2 = x ± √x2 + 1
But ey > 0 for all y and √x2 + 1 > x, therefore
ey = x + √x2 + 1
⇒ y = loge(x + √x2 + 1)
⇒ sinh−1x = log(x + √x2 + 1)
Hence, when x is real, sinh−1x is a single valued function.
Formulas and Concepts
1. tanh x = sinh x / cosh x = (ex − e−x) / (ex + e−x)
coth x = cosh x / sinh x = (ex + e−x) / (ex − e−x)
2. sech x = 1 / cosh x = 2 / (ex + e−x)
cosech x = 1 / sinh x = 2 / (ex − e−x)
(i) cos ix = cosh x
(ii) sin ix = i sinh x
(iii) cosh ix = cos x
(iv) sinh ix = i sin x
3. sinh 2x = 2 sinh x cosh x
4. cosh 2x = 2 cosh2 x − 1
5. 1 + cosh 2x = 2 cosh2 x
6. cosh 2x = 1 + 2 sinh2 x
7. 1 − cosh 2x = −2 sinh2 x
8. tanh 2x = (2 tanh x) / (1 + tanh2 x)
9. sinh 2x = (2 tanh x) / (1 − tanh2 x)
10. cosh 2x = (1 + tanh²x) / (1 - tanh²x)
11. sinh 3x = 3 sinhx + 4 sinh³x
12. cosh 3x = 4 cosh³x - 3 coshx
13. tanh 3x = (3 tanhx + tanh³x) / (1 + 3 tanh²x)
14. cos²h x - sin²h x = 1
15. cos²h x + sin²h x = cosh 2x
16. 1 - tanh²x = sech²x
17. 1 - coth²x = -cosech2x