SOLUTION: Find the exact value of x for which 4cosh(x) - e^-(x) = 3 Find the exact value of cosh^-¹(3) - sinh^-1(2)

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Question 1196149: Find the exact value of x for which 4cosh(x) - e^-(x) = 3
Find the exact value of cosh^-¹(3) - sinh^-1(2)
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
............................use identity








Rewrite the equation with:



solutions for :
if => =>
if =>

Substitute back , solve for

....take natural log of both sides





or






verify solutions:



=> true
or









Use the following identities:



so we have


=
=-> exact solution





Answer by ikleyn(52782)   (Show Source): You can put this solution on YOUR website!
.
(a) Find the exact value of x for which 4cosh(x) - e^-(x) = 3

(b) Find the exact value of cosh^-¹(3) - sinh^-1(2)
~~~~~~~~~~~~~~~~~~~


            There are two problems in the post, which is prohibited by the rules of the forum.

            So,  I will solve part  (a)  ONLY,  in order for do not transform my response into a mess. 


By the definition,  cosh(x) = .


So, I will introduce new variable  t = .  


Then the equation takes the form

     -  = 3.


Simplify it step by step

     -  = 3

    2t +  = 3

    2t^2 + 1 = 3t

    2t^2 - 3t + 1 = 0.


Apply the quadratic formula.  It will give two roots:  t = 1  and  t = .


If t = 1,  then   = 1  and  x = 0.

If t = ,  then   =   implies  x =  = -ln(2).


Thus part (a) has two solutions  x = 0  and  x = -ln(2).    ANSWER

Part (a) is just solved.

-----------------

Please do not submit more then one problem per post.

Post the problems separately.



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