Hyperbolic Functions

Question 1

What are the defintions for sinh(x) and cosh(x)?

Answer 1

sinh(x) = 0.5(e^x - e^-x)
cosh(x) = 0.5(^x + e^-x)

Question 2

Hyperbolic functions and trig funtions are defined in the same way, how is tanh cosech sech and coth defined?

Answer 2

tanh(x) = sinh(x)/cosh(x)
cosech(x) = 1/sinh(x)
sech(x) = 1/cosh(x)
coth(x) = 1/tanh(x)

Question 3

What are the graphs of cosh(x) sinh(x) and tanh(x) and there domains and ranges?

Answer 3

cosh(x): quadratic where the minimum goes through the x axis at 1 and the range is y ≥ 1

sinh(x): is like a sin^-1(x) graph but stretched where the range is all real numbers

tanh(x):is an s graph with asymptotes at y=1 and y=-1 where the range is 1≥y≥-1

The domain of all these graphs is all real numbers

Question 4

What are the inverse hyperbolic functions of sinh cosh and tanh and there domains?

Answer 4

sinh(x): arsinh(x)
cosh(x): arcosh(x) x≥1
artanh(x):|x|< 1

Question 5

What are the steps for finding inverse hyperbolic functions of sinh and cosh in terms of natural logarithms?

Answer 5

• y = arcosh(x) or arsinh(x)
• x = sinh(y) or cosh(y)
• Replace the hyperbolic side with the exponential definition
• Rearrange the expression until you have a quadratic in terms of e^y where the coefficients are in terms of x
• Complete the square rearrange for e^y =
• ln both sides are rearrange for y=

Question 6

What are the steps for finding the inverse hyperbolic function of tanh in terms of natural logarithms?

Answer 6

• y = artanh(x) so x = tanh(y)
• x = sinh(y)/cosh(y)
• use the exponential definitions to replace the top and bottom and cancel the over 2
• multiply up the bottom of the fraction and multiply both sides by e^y
• Rearrange for an expression in terms of (e^y)² then factorise the (e^y)²
• divide over the x terms to one side then ln both sides
• rearrange for y=

Question 7

What are the formulas for inverse hyperbolic functions given in the formula book? arsinh arcosh and artanh

Answer 7

arsinh(x) = ln(x+(x²+1)^1/2)
arcosh(x) = ln(x±(x²-1)^1/2) x≥1
artanh(x) = 0.5ln(1+x/1-x) |x|< 1

Question 8

How can you prove hyperbolic identities?

Answer 8

• Pick a side then using the exponential definitions replace the hyperbolic terms
• Rearrange till you have the other side

Note as with all proofs it generally helps to pick the more complex side

Question 9

How can you apply osbourne’s rule?

Answer 9

Given a trig identity you can convert to a hyperbolic identity using osbourne’s rule

• Swap all trig terms with corresponding hyperbolic terms
• Wherever you have a sin² or an implied sin² like a tan² multiply the new hyperbolic term by -1

Question 10

What hyperbolic identities are given in the formula book?

Answer 10

cosh²(x) - sinh²(x) = 1
sinh(2x) = 2sinh(x)cosh(x)
cosh(2x) = cosh²(x) + sinh²(x)

Question 11

If you are given the value of one hyperbolic function how can you find the value of another?

Answer 11

Using the identities sub in the value of one of the functions and solve for the other as long as there is only one unknown

Question 12

How can you solve hyperbolic equations?

Answer 12

• By swapping the hyperbolics with there exponential definitions and solving for x
• By using hyperbolic identities and solving as you would a trig question
• By doing a hidden quadratic
• Potentially by doing R-α

Question 13

What are the derivatives of cosh(x) sinh(x) and tanh(x) which are in the formula book?

Answer 13

d/dx sinh(x) = cosh(x)
d/dx cosh(x) = sinh(x)
d/dx tanh(x) = sech²(x)

Note must do chain rule aswell

Question 14

How can you prove the derivatives of cosh(x) and sinh(x)?

Answer 14

• Using the exponential definitions replace the hyperbolic terms
• Then differentiate e
• Then replace the exponential derivative with the hyperbolic equivalent

Question 15

How can you prove the derivative of tanh(x)?

Answer 15

• tanh(x) = sinh(x) / cosh(x)
• d/dx tanh(x) = d/dx (sinh(x) / cosh(x))
• Using the quotient rule for differentiation
• Rearrange and simplify until your answer is sech²(x)

Question 16

What are the derivatives of arsinh(x), arcosh(x) and artanh(x) that are given in the formula book?

Answer 16

d/dx arsinh(x) = 1 / (x² + 1)^1/2

d/dx arcosh(x) = 1 / (x² - 1)^1/2 x>1

d/dx arsinh(x) = 1 / 1-x² |x|<1

Note must do chain rule aswell

Question 17

What are the steps for proving the derivative of arsinh(x)?

Question 18

What are the steps for proving the derivative of arcosh(x)?

Question 19

What are the steps for proving the derivative of artanh(x)?

Question 20

What are the integrals of sinh(x) cosh(x) and tanh(x) that are given in the formula book?

Answer 20

∫sinh(x) dx = cosh(x) + c

∫cosh(x) dx = sinh(x) + c

∫tanh(x) dx = lncosh(x) + c

Note: must do reverse linearity rule aswell

Question 21

When doing integration and differentiation of hyperbolics what mustn’t you forget?

Answer 21

To apply the reverse chain rule and the chain rule

Note: don’t forget the inside of the function must be linear for integration

Question 22

What is the integration trick used to integrate composite functions where the inside of the function isn’t linear?

Answer 22

• If the outside of the bracket is a multiple of the derivative of inside the bracket
• Make the outside function = the derivative of the inside of the other function
• Ignore the outside function and integrate the other function as normal using the reverse bracket rule ignoring the reverse linearity rule

Question 23

When do you use a hyperbolic substitution to integrate?

Answer 23

• When it tells you you must use one
• When you are proving
• ∫1/ (a²±x²)
• Any integral of a square root typically in the form (x±a²) this can be one over or not

Question 24

What are the steps of integration using a hyperbolic substitution?

Answer 24

• Pick your u term i.e u = arcosh(x/a)
• Write your x term i.e x = acosh(u)
• Differentiate your x term i.e dx/du = asinh(u)
• Replace dx in the integral and replace the x in the integral
• Factorise the bottom to get either (sinh²+1) or (cosh²+1) then use the identity, cosh²(u) - sinh²(u) = 1 to replace this term
• Simplify, swap your x limits for u limits and solve

Question 25

How do you pick your ‘u’ term in integration via a hyperbolic substitution?

Answer 25

If you have a term with ((bx)² - a²)^1/2 use u = arcosh(x/(a/b))

If you have a term with ((bx)² + a²)^1/2 use u = arsinh(x/(a/b))

Question 26

When doing a hyperbolic substitution what must you keep in mind?

Answer 26

Whatever is inside the square root in your integral must have distinct x and a terms.

i.e if you have 4(x+1)² inside the square root you must absorb the 4. Or if you have a quadratic inside the square root you must complete the square

Question 26

What are the two formulas for integrating 1/(x²±a²)

Answer 26

∫1/(a²+x²)^1/2 dx = arsinh(x/a) + c

∫1/(x²-a²)^1/2 dx = arcosh(x/a) + c

Note for both of these formulas you must then apply the reverse chain rule

Note these two formulas can be proved by using the hyperbolic substitutions