I want to find out the angle for the expression a 3 + b 3 = c 3 .

Question

I want to find out the angle for the expression       a    3    +      b    3    =      c    3  .

kamphundg4 · Accepted Answer

Step 1There 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 θ is the angle opposite the side of length c, then       c    2    =      a    2    +      b    2    −  2  a  b  cos  ⁡  θ, so   cos  ⁡  θ  =                    a        2            +              b        2            −              c        2                    2      a      b          ..Now let’s look at just a few examples. If   a  =  b  =  1, then   c  =      2    3  , and   cos  ⁡  θ  =            2      −              2                  2                      /                    3                      2    ≈  0.20630    ..Step 2If   a  =  1 and   b  =  2, then   c  =      9    3  , and   cos  ⁡  θ  =            5      −              9                  2                      /                    3                      4    ≈  0.16831    ..If   a  =  1 and   b  =  3, then   c  =      28    3  , and   cos  ⁡  θ  =            10      −              28                  2                      /                    3                      6    ≈  0.12985    ..As you can see, these values of   cos  ⁡  θ are all different, so the angles themselves are also different.

suchonosdy · Accepted Answer

Step 1You 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  ..Step 2In 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.

I want to find out the angle for the expression a^3+b^3=c^3.

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