Instead of using napier's analogies, is there any simpler alternate solutions/formulas used in solving oblique spherical triangles, especially for cases 3-5?
Simpler than what method? What are cases 3-5?
amazing. I didn't think anyone did spherical trig any more.
simpler than napier's analogies...?
Yes, there are simpler alternative solutions/formulas that can be used to solve oblique spherical triangles, particularly for cases 3-5. One such alternate solution is the Law of Cosines for Spherical Triangles.
In case 3, where you are given the three sides of the triangle, you can use the Law of Cosines for Spherical Triangles to find the unknown angles. The formula is as follows:
cos(a) = cos(b) * cos(c) + sin(b) * sin(c) * cos(A)
Here, a, b, and c represent the sides of the triangle, and A represents the angle opposite the side 'a'. Once you find the values of the angles, you can use the Law of Sines for Spherical Triangles to find the remaining side lengths.
In case 4, where you are given two sides and an included angle, you can again use the Law of Cosines for Spherical Triangles to find the remaining angles. The formula is similar to the previous one, but it allows you to directly find the angle opposite the unknown side. After finding the angles, you can apply the Law of Sines to find the missing side lengths.
In case 5, where you are given two angles and an included side, you can use the Law of Sines for Spherical Triangles to find the remaining angles. The formula is as follows:
sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c)
Here, A, B, and C represent the angles of the triangle, and a, b, and c represent the side lengths. Once you find the angles, you can again use the Law of Sines or the Law of Cosines to find the missing side lengths.
These alternate solutions can often simplify the calculations involved in solving oblique spherical triangles, especially for cases 3-5.