taking this overf rom the mash discussion channel
#hyperbolic
quiet abyss
coshxcoshy=9
sinhxsinhy=8
get y=f(x)
so I did 9-8 for cosh(x-y) = 1 => x=y
but then someone said 'well cosh is even so there is another solution' which we found from doing
9+8=cosh(x+y) so y = arccosh17-x
I have two questions:
- why are there two solutions, like, evenness sounds intelligent to me, i get what evenness is, but i dont nderstand, why its not just one solution
- is there a solution method such that both answers pop up? rather than needing to do both 9-8 and 9+8 which makes no sense to me.
When i think ab the exponentials, I dont see how a +- would arise
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cobalt salmon
,w plot y=coshx
distant brookBOT
cobalt salmon
When i think ab the exponentials, I dont see how a +- would arise
It’s a many-to-one function
,w arcosh3
distant brookBOT
cobalt salmon
,w (arcosh17)/2
distant brookBOT
cobalt salmon
1. why are there two solutions, like, evenness sounds intelligent to me, i get w...
After getting x=y u can sub back in to get cosh^2(x) = 9 and u get the same solns
,w arsinh(sqrt8)
distant brookBOT
cobalt salmon
Or similarly with the other eqn
quiet abyss
ok so the algebra works
but im still not
convincedd
like
im used to sim equs where, u sub it back in, u get THE solutionm
i suppose theres no reason it mus tbe that way
idk
anyone have an intelligent way i can understand better?
violet gulch
the question seems a bit weird, it would make more sense if it said to solve the equation, rather than finding y in terms of x
cobalt salmon
im used to sim equs where, u sub it back in, u get THE solutionm
U do get both at once if u do +-
quiet abyss
U do get both at once if u do +-
wdym
like x-->-x via evennes?
cosh(-x)cosh(y)-sinh(-x)sinhy
coshxcoshy+sinhxsinhy
cosh(x+y)=1?
x=-y
wait
what is this dark magic
oh its bcs x-->-x
im stopeed dont midn me
quiet abyss
U do get both at once if u do +-
wdym
cobalt salmon
wdym
Like when u wrote down coshx = +-3
Then x = y = +-arcosh3
Or is it arcosh+-3
quiet abyss
Like when u wrote down coshx = +-3
rej. coshx=-3 => coshx=3
quiet abyss
coshx=3
e^x+e^-x=6
let U=e^x
U^2+1=6U
U^2-6U+1=0
discrim. = 32 => two solutions for x => two solutions where y=x
cobalt salmon
Ye
But i think whenever u solve an equation like coshx = … u should do +-arcosh… like with x^2 = … u take +-root
violet gulch
or u can do +- within the ln
cobalt salmon
ya
tired plover
icl I think the textbook mentions when there is 2 like coshx = k and k > 1?
something like that
if you wanna know the reasoning behind it then yeah idk 😭
distant brookBOT
tired plover
maybe to do with the graph like if you want coshx = 3 3 intersects at 2 points but cosh x = 1 intersects 1 and anything below is undefined?
cobalt salmon
Bean
quiet abyss
❤️