Is sin(10) Irrational? Proving the Irrationality of sin(10) in Degrees

  • Thread starter Thread starter SrEstroncio
  • Start date Start date
  • Tags Tags
    Irrational Proof
SrEstroncio
Messages
60
Reaction score
0

Homework Statement


Prove \sin{10}, in degrees, is irrational.


Homework Equations


None, got the problem as is.


The Attempt at a Solution


Im kinda lost.
 
Physics news on Phys.org
SrEstroncio said:

Homework Statement


Prove \sin{10}, in degrees, is irrational.


Homework Equations


None, got the problem as is.


The Attempt at a Solution


Im kinda lost.

Okay, are you familiar with proof by contradiction?

Start by contradicting the statement :

Suppose that sin(10) is irrational.
 
SrEstroncio said:

Homework Statement


Prove \sin{10}, in degrees, is irrational.


Homework Equations


None, got the problem as is.


The Attempt at a Solution


Im kinda lost.

Let x=sin(10). You'll want to find a polynomial x satisfies and show it has no rational root. Hint: sin(30)=1/2=sin(10+10+10).
 
I should suppose sin(10) is rational, if i am to contradict the statement, shouldn't i?
 
yes and that means that sin(10) can be represented by the ratio of two integers a and b sin(10) = a/b

next step...
 
SrEstroncio said:
I should suppose sin(10) is rational, if i am to contradict the statement, shouldn't i?

Zondrina's hint contains nothing of any real substance to start solving the problem. Mine does. I suggest you start trying it.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Proving something is irrational
Replies
8
Views
2K
If a, b are irrational, then is ##a^b## irrational?
Replies
15
Views
5K
Is the sum of a rational number and an irrational number always irrational?
Replies
8
Views
2K
Showing that tan(1) is irrational
Replies
4
Views
4K
Sum of a rational number and an irrational number ....
Replies
2
Views
3K
Prove that a^(1/n) is an integer or is irrational
Replies
7
Views
2K
Can Irrational Numbers Be Found Between Any Two Real Numbers?
Replies
8
Views
2K
Does an Open Cover of Rationals in [0,1] Necessarily Encompass All Irrationals?
Replies
10
Views
2K
Jordan measure on the irrationals
Replies
7
Views
2K
Square Root of an Irrational Number is Irrational
Replies
26
Views
20K

Hot Threads

Recent Insights

Back
Top