Chapter 5: Everything about Hyperbolic Functions

Question 1

Define the following hyperbolic function:

sinh x

Answer 1

(e^x-e^-x)/2

Question 2

Define the following hyperbolic function:

cosh x

Answer 2

(e^x+e^-x)/2

Question 3

Define the following hyperbolic function:

tanh x

Answer 3

(sinh x)/(cosh x)

Question 4

Define the following hyperbolic function:

csch x

Answer 4

1/(sinh x)

Question 5

Define the following hyperbolic function:

sech x

Answer 5

1/(cosh x)

Question 6

Define the following hyperbolic function:

coth x

Answer 6

(cosh x)/(sinh x)

Question 7

Complete the following hyperbolic identity:

sinh(-x) =

Answer 7

-sinh(x)

Question 8

Complete the following hyperbolic identity:

1 =

(using cosh and sinh)

Answer 8

cosh²(x)-sinh²(x)

Question 9

Complete the following hyperbolic identity:

cosh(-x) =

Answer 9

cosh(x)

Question 10

Complete the following hyperbolic identity:

1 - ? = sech²x

Answer 10

? = tanh²(x)

(Can be derived by dividing cosh²x from both sides of identity: cosh²(x)-sinh²(x)=1)

Question 11

Complete the following hyperbolic identity:

sinh(x+y) =

Answer 11

sinh(x)*cosh(y) + cosh(x)*sinh(y)

Question 12

Complete the following hyperbolic identity:

cosh(x+y) =

Answer 12

cosh(x)*cosh(y) + sinh(x)*sinh(y)

Question 13

Find the derivative of the following hyperbolic function:

sinh(x)

Answer 13

cosh(x)

Question 14

Find the derivative of the following hyperbolic function:

cosh(x)

Answer 14

sinh(x)

Question 15

Find the derivative of the following hyperbolic function:

tanh(x)

Answer 15

sech²(x)

Question 16

Find the derivative of the following hyperbolic function:

csch(x)

Answer 16

-csch(x)*coth(x)

Question 17

Find the derivative of the following hyperbolic function:

sech(x)

Answer 17

-sech(x)*tanh(x)

Question 18

Find the derivative of the following hyperbolic function:

coth(x)

Answer 18

-csch²(x)

Question 19

Find the derivative of the following inverse hyperbolic function:

sinh^-1(x)

Answer 19

1/sqrt(1+x²)

Question 20

Find the derivative of the following inverse hyperbolic function:

cosh^-1(x)

Answer 20

1/sqrt(x²-1)

Question 21

Find the derivative of the following inverse hyperbolic function:

tanh^-1(x)

Answer 21

1/(1-x²)

Question 22

Find the derivative of the following inverse hyperbolic function:

csch^-1(x)

Answer 22

-1/(|x|sqrt(x²+1))

Question 23

Find the derivative of the following inverse hyperbolic function:

sech^-1(x)

Answer 23

-1/(x*sqrt(1-x²))

Question 24

Find the derivative of the following inverse hyperbolic function:

coth^-1(x)

Answer 24

1/(1-x²)