like in pythagorean theorem angle comes 90 degree for the expression $a^2 + b^2 = c^2$, however I know that no integer solution is possible.
I want to find out the angle for the expression $a^3 + b^3 = c^3$.
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0Are you perhaps wondering if there is a form of non-euclidian geometry which has a cube law for the sides of "right-angled" triangles appropriately defined? – 2012-08-19
4 Answers
There is no single angle corresponding to the relationship $a^3+b^3=c^3$.
Suppose that a triangle has sides of lengths $a,b$, and $c$ such that $a^3+b^3=c^3$. We know from the law of cosines that if $\theta$ is the angle opposite the side of length $c$, then $c^2=a^2+b^2-2ab\cos\theta$, so
$\cos\theta=\frac{a^2+b^2-c^2}{2ab}\;.$
Now let’s look at just a few examples. If $a=b=1$, then $c=\sqrt[3]2$, and $\cos\theta=\frac{2-2^{2/3}}2\approx0.20630\;.$
If $a=1$ and $b=2$, then $c=\sqrt[3]9$, and $\cos\theta=\frac{5-9^{2/3}}4\approx0.16831\;.$
If $a=1$ and $b=3$, then $c=\sqrt[3]{28}$, and $\cos\theta=\frac{10-28^{2/3}}6\approx0.12985\;.$
As you can see, these values of $\cos\theta$ are all different, so the angles themselves are also different.
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0M.Scott : Thanks, I got your point and proof is really satisfying me – 2012-08-20
And here's a plot of the angle:
The minimum angle seems to be at $a = b$, where
$\cos\theta = 1 - \frac{1}{2^{1/3}}$
You do get slightly different answers for the Pythagorean Theorem on the surface of a sphere of radius $1,$ or the hyperbolic plane of curvature $-1.$ On the sphere, a right triangle with geodesic lengths of legs $a,b$ and hypotenuse $c$ obeys $ \cos c = \cos a \; \cos b, $ while in the hyperbolic plane with curvature $-1$ it becomes $ \cosh c = \cosh a \; \cosh b. $ In both cases, if you write out the functions as power series in $a,b,c,$ you see that the limit as $a,b,c$ all shrink to nearly $0$ is the traditional Pythagorean Theorem.
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1@WillJagy I just thought it was an acronym and was wondering what it stood for. – 2012-08-19
Expanding on Brian M. Scott's answer, since $\cos\theta=\frac{a^2+b^2-c^2}{2ab}$ and $c^3 = a^3+b^3$, $\cos\theta=\frac{a^2+b^2-(a^3+b^3)^{2/3}}{2ab} = \frac{1+(b/a)^2-(1+(b/a)^3)^{2/3}}{2} =\frac{1+r^2-(1+r^3)^{2/3}}{2} $ where $r = b/a$.
The derivative of the numerator is $2r-(2/3)(3r^2)(1+r^3)^{-1/3} = 2r - 2r^2(1+r^3)^{-1/3}$ which is never 0 (else $(1+r^3)^{1/3} = r$) so is always positive (since its value at 1 is $2-2/2^{1/3} > 0$).
For large r, $(1+r^3)^{2/3} = r^2 (1+r^{-3})^{2/3} \approx r^2(1 + (2/3)r^{-3}) = r^2 + 2/(3r) $ so $\cos\theta \approx \frac{1+r^2 - (r^2 + 2/(3r))}{2} = \frac{1-2/(3r)}{2} $ which tends to 1/2 for large $r$.
This may have an error, since I would expect it to go to 1, but I have to go now, so this this is it.
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0However some mistake is there, but the logic is correct for the explanation. – 2012-08-20