Spherical Pythagorean theorem - finding length of longer side

[Plutoniummatt]

Picture of the problem:

As seen by the diagram above, a2 < a1

But the spherical Pythagorean theorem states that cos c = (cos a)(cos b).

The triangle can either have a1,b,c or a2,b,c as its sides, which means the above equation contradicts itself. Am I missing something?

thanks.

 

[tiny-tim]

Hi Plutoniummatt!

But the spherical Pythagorean theorem states that cos c = (cos a)(cos b).

he he

a₁ isn't part of a great circle ​

 

[Plutoniummatt]

hehe thanks, this is what happens when I look up a formula and not bother to read the text that goes with it :P but I assume you meant a2

 

[tiny-tim]

oops!

 

[blue_raver22]

I would first like to clarify that the spherical Pythagorean theorem applies to spherical triangles, which are different from the traditional triangles we are used to dealing with in Euclidean geometry. In spherical geometry, the sum of the angles in a triangle can be greater than 180 degrees, and the sides are measured along the surface of a sphere rather than in a flat plane. Therefore, the traditional Pythagorean theorem does not hold true for spherical triangles.

In the case of the problem presented, the equation cos c = (cos a)(cos b) is used to find the length of the longer side, which is not necessarily the hypotenuse as it is in traditional triangles. It is important to note that this equation only applies to right spherical triangles, where one of the angles is a right angle. In this case, the longer side would be the hypotenuse, and the equation would hold true.

However, in the diagram provided, a1 and a2 are both considered the "longer side" depending on which triangle we are looking at. This leads to the contradiction you mentioned. To accurately solve for the length of the longer side in this case, we would need to know the angles of the triangle and use the appropriate formula for spherical triangles, such as the Law of Cosines.

In conclusion, the spherical Pythagorean theorem should only be applied to right spherical triangles, and the longer side may not always be the hypotenuse. It is important to understand the principles of spherical geometry before applying formulas and equations.