Hyperbolic Functions Flashcards
What are the defintions for sinh(x) and cosh(x)?
sinh(x) = 0.5(ex - e-x)
cosh(x) = 0.5(x + e-x)
Hyperbolic functions and trig funtions are defined in the same way, how is tanh cosech sech and coth defined?
tanh(x) = sinh(x)/cosh(x)
cosech(x) = 1/sinh(x)
sech(x) = 1/cosh(x)
coth(x) = 1/tanh(x)
What are the graphs of cosh(x) sinh(x) and tanh(x) and there domains and ranges?
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
What are the inverse hyperbolic functions of sinh cosh and tanh and there domains?
sinh(x): arsinh(x)
cosh(x): arcosh(x) x≥1
artanh(x):|x|< 1
What are the steps for finding inverse hyperbolic functions of sinh and cosh in terms of natural logarithms?
- 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 ey where the coefficients are in terms of x
- Complete the square rearrange for ey =
- ln both sides are rearrange for y=
What are the steps for finding the inverse hyperbolic function of tanh in terms of natural logarithms?
- 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 ey
- Rearrange for an expression in terms of (ey)2 then factorise the (ey)2
- divide over the x terms to one side then ln both sides
- rearrange for y=
What are the formulas for inverse hyperbolic functions given in the formula book? arsinh arcosh and artanh
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
How can you prove hyperbolic identities?
- 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
How can you apply osbourne’s rule?
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
What hyperbolic identities are given in the formula book?
cosh2(x) - sinh2(x) = 1
sinh(2x) = 2sinh(x)cosh(x)
cosh(2x) = cosh2(x) + sinh2(x)
If you are given the value of one hyperbolic function how can you find the value of another?
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
How can you solve hyperbolic equations?
- 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
What are the derivatives of cosh(x) sinh(x) and tanh(x) which are in the formula book?
d/dx sinh(x) = cosh(x)
d/dx cosh(x) = sinh(x)
d/dx tanh(x) = sech2(x)
Note must do chain rule aswell
How can you prove the derivatives of cosh(x) and sinh(x)?
- Using the exponential definitions replace the hyperbolic terms
- Then differentiate e
- Then replace the exponential derivative with the hyperbolic equivalent
How can you prove the derivative of tanh(x)?
- 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)
What are the derivatives of arsinh(x), arcosh(x) and artanh(x) that are given in the formula book?
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
What are the steps for proving the derivative of arsinh(x)?
What are the steps for proving the derivative of arcosh(x)?
What are the steps for proving the derivative of artanh(x)?
What are the integrals of sinh(x) cosh(x) and tanh(x) that are given in the formula book?
∫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
When doing integration and differentiation of hyperbolics what mustn’t you forget?
To apply the reverse chain rule and the chain rule
Note: don’t forget the inside of the function must be linear for integration
What is the integration trick used to integrate composite functions where the inside of the function isn’t linear?
- 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
When do you use a hyperbolic substitution to integrate?
- 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
What are the steps of integration using a hyperbolic substitution?
- 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
How do you pick your ‘u’ term in integration via a hyperbolic substitution?
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))
When doing a hyperbolic substitution what must you keep in mind?
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
What are the two formulas for integrating 1/(x2±a2)
∫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
