Chapter-6-HYPERBOLIC FUNCTION-2-
6.1 Definition.
We know that parametric co-ordinates of any point on the unit circle is ; so that these functions are called circular functions and co-ordinates of any point on unit hyperbola is i.e., . It means that the relation which exists amongst and unit circle, that relation also exist amongst and unit hyperbola. Because of this reason these functions are called as Hyperbolic functions.
For any (real or complex) variable quantity x,
(1) [Read as 'hyperbolic sine x'] (2) [Read as 'hyperbolic cosine x']
(3) (4)
(5) (6)
Note :
6.2 Domain and Range of Hyperbolic Functions.
Let x is any real number
Function Domain Range
R R
R
R
R
.
6.3 Graph of Real Hyperbolic Functions.
(1)
(2)
(3)
(4)
(5)
(6)
6.4 Formulae for Hyperbolic Functions.
The following formulae can easily be established directly from above definitions
(1) Reciprocal formulae
(i) (ii) (iii)
(iv) (v)
(2) Square formulae
(i) (ii)
(iii) (iv)
(3) Expansion or Sum and difference formulae
(i) (ii)
(iii)
(4) Formulae to transform the product into sum or difference
(i) (ii)
(iii) (iv)
(v) (vi)
(vii) (viii)
(ix) (x) (xi)
(5) Trigonometric ratio of multiple of an angle
(i) =
(ii) = = =
(iii) (iv) (v)
(vi) (vii) (viii)
(6) (i) (ii) (iii)
Example: 1 is equal to
(a) (b) (c) (d)
Solution: (c) .
Example: 2 If then is equal to
(a) (b) (c) (d)
Solution: (b) = .
Example: 3 If then is equal to
(a) (b) (c) (d)
Solution: (a)
= = .
Example: 4
(a) (b) (c) (d)
Solution: (d)
Example: 5 If then
(a) (b) (c) (d)
Solution: (d)
.
6.5 Transformation of a Hyperbolic Functions.
Since,
Also,
In a similar manner we can express , ,................ in terms of other hyperbolic functions.
6.6 Expansion of Hyperbolic Functions.
(1)
(2)
(3)
The expansion of does not exist because .
6.7 Relation between Hyperbolic and Circular Functions.
We have from Euler formulae,
........(i) and ........(ii)
Adding (i) and (ii)
Subtracting (ii) from (i)
Replacing x by ix in these values, we get
Also
Similarly replacing x by ix in the definitions of and , we get
Also,
Thus, we obtain the following relations between hyperbolic and trigonometrical functions.
(1)
(2)
(3)
(4)
(5)
(6)
Important Tips
For obtaining any formula given in (5)th article, use the following substitutions in the corresponding formula for trigonometric functions.
For example,
For finding out the formula for in terms of , replace by and by by in the following formula of trigonometric function of :
we get,
6.8 Period of Hyperbolic Functions.
If for any function then is called the Periodic function and least positive value of T is called the Period of the function.
and
Therefore the period of these functions are respectively and . Also period of cosech x, sech x and are respectively , and .
Note : Remember that if the period of is T, then period of will be .
Hyperbolic function are neither periodic functions nor their curves are periodic but they show the algebraic properties of periodic functions and having imaginary period.
Example: 6 If , then A equals
(a) (b) (c) (d)
Solution: (a)
Example: 7 If then is equal to
(a) (b) (c) (d)
Solution: (c)
= =
= .
Example: 8 The value of is
(a) ¬– 1 (b) i (c) 0 (d) 1
Solution: (a) = .
Example: 9 equals
(a) (b) (c) (d)
Solution: (d)
= = = .
Example: 10 is equal to
(a) (b) (c) (d)
Solution: (b) .
Example: 11 equals
(a) (b) (c) (d)
Solution: (c) = .
Example: 12 The period of is
(a) (b) (c) (d)
Solution: (a) Since the period of is so the period of is .
Example: 13 The period of is
(a) (b) (c) (d)
Solution: (c) Since period of is , therefore period of will be .
6.9 Inverse Hyperbolic Functions.
If then y is called the inverse hyperbolic sine of x and it is written as . Similarly etc. can be defined.
(1) Domain and range of Inverse hyperbolic function
Function Domain Range
R R
[1, ) R
R
R – [–1, 1]
(0, 1] R
(2) Relation between inverse hyperbolic function and inverse circular function
Method : Let
=
Therefore we get the following relations
(i) (ii) (iii)
(iv) (v)
(3) To express any one inverse hyperbolic function in terms of the other inverse hyperbolic functions
To express in terms of the others
(i) Let
(ii)
(iii) =
(iv)
(v)
(vi) Also,
From the above, it is clear that
Note : If x is real then all the above six inverse functions are single valued.
(4) Relation between inverse hyperbolic functions and logarithmic functions
Method : Let
But and
By the above method we can obtain the following relations between inverse hyperbolic functions and principal values of logarithmic functions.
(i) (ii)
(iii) (iv)
(v) (vi)
Note : Formulae for values of and may be obtained by replacing x by in the values of and respectively.
6.10 Separation of Inverse Trigonometric and Inverse Hyperbolic Functions.
If = then , is called the inverse sine of . We can write it as,
Here the following results for inverse functions may be easily established.
(1)
(2)
=
(3) =
(4) or
(5) or
(6)
and
Since each inverse hyperbolic function can be expressed in terms of logarithmic function, therefore for separation into real and imaginary parts of inverse hyperbolic function of complex quantities use the appropriate method.
Note : Both inverse circular and inverse hyperbolic functions are many valued.
Example: 14 If then
(a) (b) (c) (d)
Solution: (c) .
Example: 15 If then
(a) 2 (b) 1 (c) 3 (d) 5
Solution: (a) =
.
Example: 16
(a) (b) (c) (d)
Solution: (b) =
.
Example: 17 is
(a) (b) (c) (d) None of these
Solution: (c) = = .
Example: 18 is
(a) (b) (c) (d)
Solution: (b) = .
Example: 19 If then
(a) (b) (c) (d) None of these
Solution: (a)
=
= = .
Example: 20 If , then the value of is
(a) (b) (c) (d)
Solution: (a) Let ;
By componendo and Dividendo rule,
.
Example: 21 The value of is
(a) (b) (c) (d)
Solution: (a) Here = = .
Example: 22 is equal to
(a) (b) (c) (d) None of these
Solution: (a) We know that,
Putting the value of
or ( )
.
Example: 23 The value of is
(a) (b) (c) (d)
Solution: (a) Let , then
.
Example: 24 If then is equal to
(a) (b) (c) (d)
Solution: (c) = =
Example: 25 is equal to
(a) (b) (c) (d)
Solution: (b) = = = .
Example: 26 If ,then the correct statement is
(a) (b) (c) (d)
Solution: (c) Given that, or or or .
Example: 27 Find real part of
(a) (2) (b) (c) (d) 0
Solution: (a) Real part = .
Example: 28 Find real part of
(a) – 1 (b) 1 (c) 0 (d) None of these
Solution: (c) We know that
= .
Example: 29 Find imaginary part of
(a) (b) (c) 0 (d) None of these
Solution: (b) = .
Example: 30 Find real part of
(a) (b) (c) log (d) None of these
Solution: (b) Expression
Where
= =
Real part = Imaginary part = .
Example: 31 Find imaginary part of
(a) (b) (c) (d) None of these
Solution: (a) Let
=
By comparing we get, ......(i) and .........(ii)
From (ii),
from (i) or =
= =
=
Real part = Imaginary part = .
***
1. The value of is
(a) (b) (c) (d) None of these
2. equals
(a) (b) (c) (d) None of these
3. Which of the following statement is true
(a) (b) (c) (d)
4. equals
(a) (b) (c) (d)
5. equals
(a) (b) (c) (d)
6. Which of the following functions is not defined at
(a) (b) (c) (d)
7. The value of is
(a) (b) (c) (d)
8. The value of is
(a) (b) (c) (d)
9. is equal to
(a) (b) (c) 1 (d) – 1
10. is equal to
(a) (b) (c) i (d) – 1
11. If then equals
(a) (b) (c) (d)
12. If then the value of is
(a) (b) (c) (d)
13. If , then the value of is
(a) (b) (c) (d)
14. ,then the value of is
(a) (b) (c) (d)
15. If then the value of is
(a) 1 (b) – 1 (c) (d)
16. equals
(a) (b) (c) (d)
17. If and , then the value of p is
(a) (b) (c) (d) None of these
18. If then is equal to
(a) (b) (c) 0 (d) 1
19. If and then equals
(a) (b) (c) (d) 1
20. If and then equals
(a) (b)
(c) (d) None of these
21. The value of is
(a) – 2 (b) 2 (c) – 2 i (d) 2 i
22. equals
(a) (b) (c) (d)
23. is equal to
(a) 0 (b) 2 (c) (d) None of these
24. equals
(a) (b) (c) (d)
25. If then A equals
(a) (b) (c) (d)
26. The imaginary part of is
(a) (b) (c) (d)
27. Imaginary part of
(a) (b) (c) (d)
28. Real part of is
(a) (b) (c) (d)
29. The value of is
(a) (b) (c) (d)
30. is equal to
(a) 1 (b) – 1 (c) (d)
31. The period of is
(a) (b) (c) (d)
32. The period of is
(a) (b) (c) (d)
33. The period of is
(a) (b) (c) (d)
34. If = , then the value of is
(a) (b) (c) (d) None of these
35. If ,then value of is
(a) (b) (c) (d)
36.
(a) (b) (c) (d) None of these
37.
(a) (b) (c) (d)
38.
(a) (b) (c) (d)
39. The value of is
(a) 0 (b) (c) (d) None of these
40. equals
(a) (b) (c) (d) None of these
41. equals
(a) (b) (c) (d)
42. The value of is
(a) (b) (c) (d)
43. equals
(a) (b) (c) (d) None of these
44. If then the value of y is
(a) 1 (b) 0 (c) (d) – 1
45. The value of is
(a) log 2 (b) log
(c) (d) None of these
46. If then K equals
(a) 1 (b) 0 (c) 2 (d) None of these
47. equals
(a) (b) (c) (d) None of these
48. is equal to
(a) (b) (c) (d)
49. If , then the value of x is
(a) (b) (c) (d)
50. is equal to
(a) (b) (c) (d)
51. is equal to
(a) (b) (c) (d)
52. is equal to
(a) (b) (c) (d)
53. If then equals
(a) (b)
(c) (d)
54. The general value of is
(a) (b) (c) (d)
55. If , then y is equal to
(a) (b) (c) (d)
56. The imaginary part of is
(a) (b) (c) (d) None of these
57. If then the equation with roots and
(a) (b) (c) (d)
58. The value of is
(a) (b) (c) (d) None of these
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
c c d b c b a c a a d b a d a c c a c a
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
c a b c b d b a b a b c c c d b d d b b
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
a d a b c c a c c a b c a b d c d a
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