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

Zoagliaj

Zoagliaj

Answered

2022-07-20

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

Answer & Explanation

kamphundg4

kamphundg4

Expert

2022-07-21Added 20 answers

Step 1
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 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 2
If 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

suchonosdy

Expert

2022-07-22Added 3 answers

Step 1
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 ..
Step 2
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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