Pure 2.6 - Hyperbolic Functions

Question 1

What is the formula for the hyperbolic sine (or sinh) function in terms of e?

Answer 1

sinh(𝑥)=(eˣ-e⁻ˣ)/2

Question 2

What is the formula for the hyperbolic cosine (or cosh) function in terms of e?

Answer 2

cosh(𝑥)=(eˣ+e⁻ˣ)/2

Question 3

What is the formula for the hyperbolic tangent (or tanh) function in terms of e?

Answer 3

sinh(𝑥)=sinh(𝑥)/cosh(𝑥)=(eˣ-e⁻ˣ)/(eˣ+e⁻ˣ)=(e₂ˣ-1)/(e₂ˣ+1)

Question 4

What are the hyperbolic functions of the reciprocal trigonometric functions?

Answer 4

cosech(𝑥)=1/sinh(𝑥)=2/(eˣ-e⁻ˣ)
sech(𝑥)=1/cosh(𝑥)=2/(eˣ+e⁻ˣ)
coth(𝑥)=1/tanh(𝑥)=(e₂ˣ+1)/(e₂ˣ-1)

Question 5

Describe the relationship between the values of sinh and -sinh.

Answer 5

For any value 𝑎, sinh(-𝑎)=-sinh(𝑎)

Question 6

Describe the relationship between the values of cosh and -cosh.

Answer 6

For any value 𝑎, cosh(-𝑎)=-cosh(𝑎)

Question 7

What is an even function?

Answer 7

Any function where f(-𝑥)=f(𝑥)

Question 8

What is an odd function?

Answer 8

Any function where f(-𝑥)=-f(𝑥)

Question 9

What are the inverse hyperbolic functions and their domains if they have any?

Answer 9

𝑦=sinh(𝑥) → 𝑦=arcinh(𝑥)
𝑦=cosh(𝑥), 𝑥≥0→ 𝑦=arcosh(𝑥), 𝑥≥1
𝑦=tanh(𝑥) → 𝑦=artanh(𝑥), |𝑥|<1

Question 10

What are the definitions of the inverse hyperbolic functions in terms of 𝑥?

Answer 10

arsinh(𝑥)=ln(𝑥+√(𝑥²+1))
arcosh(𝑥)=ln(𝑥+√(𝑥²-1)), 𝑥≥1
artanh(𝑥)=½ln((1+𝑥)/(1-𝑥)), |𝑥|<1

Question 11

What is cosh²(𝑨)-sinh²(𝑨)

Answer 11

1

Question 12

What are the hyperbolic addition formulae?

Answer 12

sinh(𝐴±𝐵)≡sinh(𝐴)cosh(𝐵)±cosh(𝐴)sinh(𝐵)

cosh(𝐴±𝐵)≡cosh(𝐴)cosh(𝐵)±sinh(𝐴)sinh(𝐵)

Question 13

Describe Osborn’s rule.

Answer 13

To get the hyperbolic identity of the corresponding trigonometric identity;
replace cos by cosh
replace sin with sinh
However replace any product or implied product of two sin terms by minus the product of two sinh terms

Question 14

f(𝑥)=sinh(𝑥) find f’(𝑥).

Answer 14

cosh(𝑥)

Question 15

f(𝑥)=cosh(𝑥) find f’(𝑥).

Answer 15

sinh(𝑥)

Question 16

f(𝑥)=tanh(𝑥) find f’(𝑥).

Answer 16

sech²(𝑥)

Question 17

f(𝑥)=arsinh(𝑥) find f’(𝑥).

Answer 17

1/√(𝑥²+1)

Question 18

f(𝑥)=arcosh(𝑥) find f’(𝑥).

Answer 18

1/√(𝑥²-1), 𝑥>1

Question 19

f(𝑥)=artanh(𝑥) find f’(𝑥).

Answer 19

1/1-𝑥², |𝑥|<1

Question 20

f’(𝑥)=sinh(𝑥) find f(𝑥).

Answer 20

cosh(𝑥)+𝑐

Question 21

f’(𝑥)=cosh(𝑥) find f(𝑥).

Answer 21

sinh(𝑥)+𝑐

Question 22

f’(𝑥)=1/√(𝑥²+1) find f(𝑥).

Answer 22

arsinh(𝑥)+𝑐

Question 23

f’(𝑥)=1/√(𝑥²-1) find f(𝑥).

Answer 23

arcosh(𝑥)+𝑐, 𝑥>1

Question 24

f’(𝑥)=tanh(𝑥) find f(𝑥).

Answer 24

ln(cosh(𝑥))+𝑐

Question 25

f’(𝑥)=1/√(𝑎²+𝑥²) find f(𝑥).

Answer 25

arsinh(𝑥𝑎/)+𝑐

Question 26

f’(𝑥)=1/√(𝑥²-𝑎²) find f(𝑥).

Answer 26

arcosh(𝑥/𝑎)+𝑐, 𝑥>𝑎