Cosine Rule: n=2,3-x,y,z Calculation

[Terry Coates]

Homework Statement

Is it significant that the angle between x and y in a triangle sides x y and z based in the equation below , is dependent on the ratio x/y when n is greater than 2 yet is always 90 degrees when n = 2?

Relevant Equations

x^n + y^n - z^n = 0

n = 3,x = 1, y = 10 z = (10^3 +9^3)^(1/3 = (1000 + 1729)^1/3
Cos (Angle xy) = (x^2 +y^2-z^2)/(2 x.y)
n = 2,x= 3, y=4 z = (3^2 +4^2)^0.5 = 5
Cos (Angle xy) = (3^2 +4^2 -5^2)/(2.3.4) = cos (0) = 1

 

[Mark44]

Homework Statement:: Is it significant that the angle between x and y in a triangle sides x y and z based in the equation below , is dependent on the ratio x/y when n is greater than 2 yet is always 90 degrees when n = 2?
Relevant Equations:: x^n + y^n - z^n = 0

n = 3,x = 1, y = 10 z = (10^3 +9^3)^(1/3 = (1000 + 1729)^1/3
Cos (Angle xy) = (x^2 +y^2-z^2)/(2 x.y)
n = 2,x= 3, y=4 z = (3^2 +4^2)^0.5 = 5
Cos (Angle xy) = (3^2 +4^2 -5^2)/(2.3.4) = cos (0) = 1

I don't understand what you're asking. In the Cosine Rule (AKA Law of Cosines), the exponent is fixed at 2. When you change the exponent to 3 or higher, you're no longer dealing with the Law of Cosines or triangles.
In your second example, the triangle is a 3-4-5 right triangle, so naturally the cosine of the right angle is 0.

 

[Charles Link]

Usually x,y, and z are perpendicular coordinates, and you don't make a triangle out of their lengths. If you attempt to, then in most cases, the x and y are no longer perpendicular. (I think you found the one case, $n=2$, for which they are perpendicular). You also need to check that $z<x+y$. (Perhaps you already did).

 

[Terry Coates]

Usually x,y, and z are perpendicular coordinates, and you don't make a triangle out of their lengths. If you attempt to, then in most cases, the x and y are no longer perpendicular. (I think you found the one case, $n=2$, for which they are perpendicular). You also need to check that $z<x+y$. (Perhaps you already did).

Thanks. Rather a silly question. It came about because I have been comparing what applies with a power higher than 2, to what applies with a modular power 2 then for instance, you get irrational sides to a triangle. also you get an oval curve instead of an ellipse. Nearest to an ellipse with power 5.