How can I simplify the exact value of arcsin(4/5)=x?

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Homework Statement

I got down to here...

x = i ln( 5/(4i + 3) ) + 2 pi n

n set of integers positive and negative

ok I was wondering if I could simplify this some more... i.e. I checked my answer in my calculator it was right just wondering that maybe...

x = i ln( i^((-i 2 ln 5)/pi) / ( i^((-i 2 ln 4)/pi + 1) + i ^((-i 2 ln 3)/pi) ) )+ 2 pi n

my gut tells me that it can be simplified some way or another if you turn all of the real numbers into complex numbers like with all numbers with the same base... but I can't seem to simplify... it's really bothering me as my gut tells me that there is a way as any fraction for the most part that have the same base with different exponents can be simplified...

like if the equation was

5^6/(5^2 + 5^8)

could be simplifed so... I feel dumb I can't simplify this one further =(

Homework Equations

The Attempt at a Solution

 

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Just in case you guys are working on the same summer assignment... lol this is Dylan H. by the way... just incase teacher googles my work :O... it's been done before =(... no teacher that was me lol

 

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ok well let's see
A^B + A^C = A^D(A^(B-D) + A^(C-D)
hmmm... take out D from both... cancel it out... but what is D... :O

 

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Can someone help me siplify it further

 

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x = i ln( i^((-i 2 ln 5)/pi) / ( i^((-i 2 ln 4)/pi + 1) + i ^((-i 2 ln 3)/pi) ) )+ 2 pi n

can i sipmlify the part in the natural log further with all of the exponents witin the natural log my gut tells me i can becasue they are now all in the same base... just can't figure out how to

 

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does somebody know how to simplify this further or if it's even possible?

 

[Mark44]

Homework Statement

I got down to here...

x = i ln( 5/(4i + 3) ) + 2 pi n

What's the question? Is it to find the exact value of x = arcsin(4/5)? What did you do to rewrite the equation as x = i ln( 5/(4i + 3) ) + 2 pi n? How is that simpler than arcsin(4/5)?

n set of integers positive and negative

ok I was wondering if I could simplify this some more... i.e. I checked my answer in my calculator it was right just wondering that maybe...

x = i ln( i^((-i 2 ln 5)/pi) / ( i^((-i 2 ln 4)/pi + 1) + i ^((-i 2 ln 3)/pi) ) )+ 2 pi n

my gut tells me that it can be simplified some way or another if you turn all of the real numbers into complex numbers like with all numbers with the same base... but I can't seem to simplify... it's really bothering me as my gut tells me that there is a way as any fraction for the most part that have the same base with different exponents can be simplified...

like if the equation was

5^6/(5^2 + 5^8)

could be simplifed so... I feel dumb I can't simplify this one further =(

Homework Equations

The Attempt at a Solution

 

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it's the exact value of the solution
arcisn(4/5)=x
I solved for x using eluer's formulas for sin and what not put it in your calculator and put in arcsin(4/5) it's right

I rewrote it all in terms of i and was wondering how do I simplify

 

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it's not that it's simpler it's just the exact value instead of using
.927295218 radians as it's wrong and only a appoximation one can find the exact value as I have done I then rewrote it all of it using complex numbers and now that everything has the same base I was wondering if somebody could help me simplify

 

[Mark44]

Why not leave it at arcsin(4/5)? I don't see any advantage in writing it in terms of logs and imaginary numbers.

 

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I was told to "evalutate" nomrally that means what um to set equal to something other than itself?

like I could just put on every single question as the answer itself... it's not wrong of course but normally evaluate means not using the idenity property or whatever were x=x etc... no? to express the question in a different way?

I think my anser is correct no like

arcsin(4/5) does equal arcsin(4/5) but can you do that in your head I can't so I found the exact value... some people can evaluate the answer I got in their heads to get a numerical value... arcsin(4/5) really is difficult to get a numerical value from...

l don't know you bring up a good point but I think I'm justified in writing that for my answer

 

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Anyways =O can you help my simplify

 

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to say that cannot be siplified furhter bothers me...

 

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much like say what's
2+1 and you put 2+1
it's deffinatly not wrong lol but normally evaluate means express without using the identiy property what ever one that is lol

but you I would like to simplify the expression above further... I was wondering if maybe

ln(pi)/pi could be simplifed some how...

 

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It's the reason I chose not to factor out the -i2 in the second expression I believe I can simplify ln(pi)/pi somehow also the fraction in the first term that is terms of the same base... to say cannot be simplifed... not buying it lol if it was

5^5/(5^2 + 5^6)

could be simplifed sense it's all in terms of the same base I just don't know how to simplify my answer...

 

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don't know if it really helps but

ln(pi) = ln(-2 i ln(i) )

 

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to say that cannot be siplified furhter bothers me...

I just realized that I forgot the 2 in the denomentor of the natural log

should be i^((-2ln4)/pi + 1) not i^((-iln4)/pi + 1)

 

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Well I'm trying to simplify and don't udnerstand why this isn't correct

i^(-i2)
does not equal
-1^-i
did i^2 = -1

like if the question was
5^10
I could do
5^(2*5)
which does equal
25^5

Can somone explain why I can't do this with complex numbers?

 

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umm...
e = i^((-2 i)/pi)

 

[Mark44]

I was told to "evalutate" nomrally that means what um to set equal to something other than itself?

like I could just put on every single question as the answer itself... it's not wrong of course but normally evaluate means not using the idenity property or whatever were x=x etc... no? to express the question in a different way?

I think my anser is correct no like

arcsin(4/5) does equal arcsin(4/5) but can you do that in your head I can't so I found the exact value... some people can evaluate the answer I got in their heads to get a numerical value... arcsin(4/5) really is difficult to get a numerical value from...

l don't know you bring up a good point but I think I'm justified in writing that for my answer

"To evaluate" just means find the value of. It would help if you gave the exact wording of the problem. If all you need to do is evaluate arcsin(4/5), you don't need all that work you show. arcsin(4/5) is about 53.13 degrees.