THE RIGHT ANGLED SPHERICAL TRIANGLE

Did you know that all spherical triangles can be solved using only formulae using the right angled spherical triangle?

Yes, you can divide up any spherical triangle into two right angled spherical triangles, just as you can with plane triangles.

Furthermore, given any two known parameters of a right angled spherical triangle, you can find all the other three parameters with formulae no more complicated than the product or division of the trigonometric functions of the two known parameters. The table below gives all 30 possible formulae that you could ever need. Only sin, cos, tan and cot are used. The formulae have been computer-checked.

A,B,C are the vertice angles, with C as the right angle.
a,b,c are the angles on the opposite sides.

So there are five parameters with two degrees of freedom. From the table below, select first the unknown quantity from the left hand side, then select which formula contains the two known parameters.

1 cos(A) = sin(B) * cos(a)
2 cos(A) = tan(b) / tan(c)
3 sin(A) = cos(B) / cos(b)
4 sin(A) = sin(a) / sin(c)
5 tan(A) = cot(B) / cos(c)
6 tan(A) = tan(a) / sin(b)

7 cos(B) = sin(A) * cos(b)
8 cos(B) = tan(a) / tan(c)
9 sin(B) = cos(A) / cos(a)
10 sin(B) = sin(b) / sin(c)
11 tan(B) = cot(A) / cos(c)
12 tan(B) = tan(b) / sin(a)

13 cos(a) = cos(A) / sin(B)
14 cos(a) = cos(c) / cos(b)
15 sin(a) = tan(b) / tan(B)
16 sin(a) = sin(A) * sin(c)
17 tan(a) = cos(B) * tan(c)
18 tan(a) = tan(A) * sin(b)

19 cos(b) = cos(B) / sin(A)
20 cos(b) = cos(c) / cos(a)
21 sin(b) = tan(a) / tan(A)
22 sin(b) = sin(B) * sin(c)
23 tan(b) = cos(A) * tan(c)
24 tan(b) = tan(B) * sin(a)

25 cos(c) = cot(A) * cot(B)
26 cos(c) = cos(a) * cos(b)
27 sin(c) = sin(a) / sin(A)
28 sin(c) = sin(b) / sin(B)
29 tan(c) = tan(b) / cos(A)
30 tan(c) = tan(a) / cos(B)

It would be nice to know that this table is useful or interesting to anyone. Please let me have your comments. Thank you.

davidb67$clara.co.uk