It is true that $2sin(54)$ equals $(1+sqrt5)/2$?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












From a popular math site I saw the formula of phi number which is $2 sin(54^circ )$. But we also know it is $(1+sqrt5)/2$ . With calculator this equation has problems. Do you think $2sin(54)$ is just approximation?
By the way is $sin(54^circ)$ irrational?




number-theory




share|cite|improve this question





















  • No, it is not an approximation, we indeed have: $$2sin (54^circ)=varphi $$ and yes, it is irrational.
    – projectilemotion
    2 days ago















up vote
0
down vote

favorite












From a popular math site I saw the formula of phi number which is $2 sin(54^circ )$. But we also know it is $(1+sqrt5)/2$ . With calculator this equation has problems. Do you think $2sin(54)$ is just approximation?
By the way is $sin(54^circ)$ irrational?




number-theory




share|cite|improve this question





















  • No, it is not an approximation, we indeed have: $$2sin (54^circ)=varphi $$ and yes, it is irrational.
    – projectilemotion
    2 days ago













up vote
0
down vote

favorite









up vote
0
down vote

favorite











From a popular math site I saw the formula of phi number which is $2 sin(54^circ )$. But we also know it is $(1+sqrt5)/2$ . With calculator this equation has problems. Do you think $2sin(54)$ is just approximation?
By the way is $sin(54^circ)$ irrational?




number-theory




share|cite|improve this question













From a popular math site I saw the formula of phi number which is $2 sin(54^circ )$. But we also know it is $(1+sqrt5)/2$ . With calculator this equation has problems. Do you think $2sin(54)$ is just approximation?
By the way is $sin(54^circ)$ irrational?





number-theory





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago









amWhy

189k25219431




189k25219431









asked 2 days ago









Talha ŞAHİN

394




394











  • No, it is not an approximation, we indeed have: $$2sin (54^circ)=varphi $$ and yes, it is irrational.
    – projectilemotion
    2 days ago

















  • No, it is not an approximation, we indeed have: $$2sin (54^circ)=varphi $$ and yes, it is irrational.
    – projectilemotion
    2 days ago
















No, it is not an approximation, we indeed have: $$2sin (54^circ)=varphi $$ and yes, it is irrational.
– projectilemotion
2 days ago





No, it is not an approximation, we indeed have: $$2sin (54^circ)=varphi $$ and yes, it is irrational.
– projectilemotion
2 days ago











1 Answer
1






active

oldest

votes

















up vote
0
down vote













here is how you find it
enter image description here



enter image description here



now you can apply the $sin(3x)$ formula $3*sin(x)-4(sinx)^3$



yes, $sin(54)$ is irrational.



proof:



$sin(54)=(1+sqrt5)/4$
thus $sqrt5$=$4*sin(54)-1$



as you know that $sqrt5$ is irrational



also 1 is rational



thus $4*sin(54)$ has to be irrational



4 is rational and sin(54) is irrational






share|cite|improve this answer























  • Sorry, your images cannot be opened. But when I try to subtract one from the other, it must be 0, calculator says it is 2,58599e-39
    – Talha ŞAHÄ°N
    2 days ago







  • 1




    it is because the calculators represent and store the numbers as floating point
    – James
    2 days ago






  • 1




    there is always some error because we can not take the infinite length of mantissa in floating point representation this leads to error
    – James
    2 days ago










  • I am not sure why images are not opening it is either the fault of your internet connection or stack exchange try opening them in laptop or pc if you have one
    – James
    2 days ago










  • I just learned our government has denied imgur, also wiki, anyway.
    – Talha ŞAHÄ°N
    2 days ago










Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);








 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2872021%2fit-is-true-that-2-sin54-equals-1-sqrt5-2%23new-answer', 'question_page');

);

Post as a guest

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













here is how you find it
enter image description here



enter image description here



now you can apply the $sin(3x)$ formula $3*sin(x)-4(sinx)^3$



yes, $sin(54)$ is irrational.



proof:



$sin(54)=(1+sqrt5)/4$
thus $sqrt5$=$4*sin(54)-1$



as you know that $sqrt5$ is irrational



also 1 is rational



thus $4*sin(54)$ has to be irrational



4 is rational and sin(54) is irrational






share|cite|improve this answer























  • Sorry, your images cannot be opened. But when I try to subtract one from the other, it must be 0, calculator says it is 2,58599e-39
    – Talha ŞAHÄ°N
    2 days ago







  • 1




    it is because the calculators represent and store the numbers as floating point
    – James
    2 days ago






  • 1




    there is always some error because we can not take the infinite length of mantissa in floating point representation this leads to error
    – James
    2 days ago










  • I am not sure why images are not opening it is either the fault of your internet connection or stack exchange try opening them in laptop or pc if you have one
    – James
    2 days ago










  • I just learned our government has denied imgur, also wiki, anyway.
    – Talha ŞAHÄ°N
    2 days ago














up vote
0
down vote













here is how you find it
enter image description here



enter image description here



now you can apply the $sin(3x)$ formula $3*sin(x)-4(sinx)^3$



yes, $sin(54)$ is irrational.



proof:



$sin(54)=(1+sqrt5)/4$
thus $sqrt5$=$4*sin(54)-1$



as you know that $sqrt5$ is irrational



also 1 is rational



thus $4*sin(54)$ has to be irrational



4 is rational and sin(54) is irrational






share|cite|improve this answer























  • Sorry, your images cannot be opened. But when I try to subtract one from the other, it must be 0, calculator says it is 2,58599e-39
    – Talha ŞAHÄ°N
    2 days ago







  • 1




    it is because the calculators represent and store the numbers as floating point
    – James
    2 days ago






  • 1




    there is always some error because we can not take the infinite length of mantissa in floating point representation this leads to error
    – James
    2 days ago










  • I am not sure why images are not opening it is either the fault of your internet connection or stack exchange try opening them in laptop or pc if you have one
    – James
    2 days ago










  • I just learned our government has denied imgur, also wiki, anyway.
    – Talha ŞAHÄ°N
    2 days ago












up vote
0
down vote










up vote
0
down vote









here is how you find it
enter image description here



enter image description here



now you can apply the $sin(3x)$ formula $3*sin(x)-4(sinx)^3$



yes, $sin(54)$ is irrational.



proof:



$sin(54)=(1+sqrt5)/4$
thus $sqrt5$=$4*sin(54)-1$



as you know that $sqrt5$ is irrational



also 1 is rational



thus $4*sin(54)$ has to be irrational



4 is rational and sin(54) is irrational






share|cite|improve this answer















here is how you find it
enter image description here



enter image description here



now you can apply the $sin(3x)$ formula $3*sin(x)-4(sinx)^3$



yes, $sin(54)$ is irrational.



proof:



$sin(54)=(1+sqrt5)/4$
thus $sqrt5$=$4*sin(54)-1$



as you know that $sqrt5$ is irrational



also 1 is rational



thus $4*sin(54)$ has to be irrational



4 is rational and sin(54) is irrational







share|cite|improve this answer















share|cite|improve this answer



share|cite|improve this answer








edited 2 days ago


























answered 2 days ago









James

325113




325113











  • Sorry, your images cannot be opened. But when I try to subtract one from the other, it must be 0, calculator says it is 2,58599e-39
    – Talha ŞAHÄ°N
    2 days ago







  • 1




    it is because the calculators represent and store the numbers as floating point
    – James
    2 days ago






  • 1




    there is always some error because we can not take the infinite length of mantissa in floating point representation this leads to error
    – James
    2 days ago










  • I am not sure why images are not opening it is either the fault of your internet connection or stack exchange try opening them in laptop or pc if you have one
    – James
    2 days ago










  • I just learned our government has denied imgur, also wiki, anyway.
    – Talha ŞAHÄ°N
    2 days ago
















  • Sorry, your images cannot be opened. But when I try to subtract one from the other, it must be 0, calculator says it is 2,58599e-39
    – Talha ŞAHÄ°N
    2 days ago







  • 1




    it is because the calculators represent and store the numbers as floating point
    – James
    2 days ago






  • 1




    there is always some error because we can not take the infinite length of mantissa in floating point representation this leads to error
    – James
    2 days ago










  • I am not sure why images are not opening it is either the fault of your internet connection or stack exchange try opening them in laptop or pc if you have one
    – James
    2 days ago










  • I just learned our government has denied imgur, also wiki, anyway.
    – Talha ŞAHÄ°N
    2 days ago















Sorry, your images cannot be opened. But when I try to subtract one from the other, it must be 0, calculator says it is 2,58599e-39
– Talha ŞAHÄ°N
2 days ago





Sorry, your images cannot be opened. But when I try to subtract one from the other, it must be 0, calculator says it is 2,58599e-39
– Talha ŞAHÄ°N
2 days ago





1




1




it is because the calculators represent and store the numbers as floating point
– James
2 days ago




it is because the calculators represent and store the numbers as floating point
– James
2 days ago




1




1




there is always some error because we can not take the infinite length of mantissa in floating point representation this leads to error
– James
2 days ago




there is always some error because we can not take the infinite length of mantissa in floating point representation this leads to error
– James
2 days ago












I am not sure why images are not opening it is either the fault of your internet connection or stack exchange try opening them in laptop or pc if you have one
– James
2 days ago




I am not sure why images are not opening it is either the fault of your internet connection or stack exchange try opening them in laptop or pc if you have one
– James
2 days ago












I just learned our government has denied imgur, also wiki, anyway.
– Talha ŞAHÄ°N
2 days ago




I just learned our government has denied imgur, also wiki, anyway.
– Talha ŞAHÄ°N
2 days ago












 

draft saved


draft discarded


























 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2872021%2fit-is-true-that-2-sin54-equals-1-sqrt5-2%23new-answer', 'question_page');

);

Post as a guest
































































Comments

Post a Comment

Popular posts from this blog

What is the equation of a 3D cone with generalised tilt?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
Read more

Color the edges and diagonals of a regular polygon

Image
Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
Read more

Relationship between determinant of matrix and determinant of adjoint?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
Read more