# Triangular Asterism Trigonometry between Alnitak, Saiph, and Sirius?

I am using the SIMBAD database for Alnitak, Saiph, and Sirius:

Alnitak ; FK5 coord. (J2000): RA = 05 40 45.527 DEC = -01 56 33.26;
Saiph   ; FK5 coord. (J2000): RA = 05 47 45.389 DEC = -09 40 10.58;
Sirius  ; FK5 coord. (J2000): RA = 06 45 08.917 DEC = -16 42 58.02;


What is the trigonometric relationship in degrees (length of sides and inclusive angles to four decimal places) between these three stars when they are used to form a triangular aterism?

• I guess, to measure the length of sides, you should also give the distance. May 9, 2014 at 1:03
• Question is not clear. Do you want spherical triangle formulas? Or real space ones? May 9, 2014 at 6:01
• @Envite Since the OP mentions "in degrees", I'm guessing they are looking for spherical triangle formulas. en.wikipedia.org/wiki/Great-circle_distance should do the trick
– user21
Oct 8, 2014 at 21:06

This is a relatively straightforward application of spherical trigonometry. (See also: About coordinate systems and angle differences and Angular Distance Between Two Points on a Sphere)

The expression to compute the angle on the celestial sphere between to points is: $$\Psi = \arccos\left(\sin\theta_1\sin\theta_2 + \cos\theta_1\cos\theta_2\cos(\phi_1-\phi_2)\right)$$ where $$\Psi$$ is the angular separation, $$\phi_1$$ and $$\phi_2$$ are the right ascensions of the first and second direction, and $$\theta_1$$ and $$\theta_2$$ are the declinations of the first and second direction.

Star Pair Separation (deg)
Alnitak - Saiph 7.9203
Alnitak - Sirius 21.6568
Saiph - Sirius 15.6351

I am including my conversion of the RA and DEC to degrees in case I have made a mistake if anyone wants to check my computations.

Star RA (HH MM SS) RA (deg) DEC (DD MM SS) DEC (deg)
Alnitak 05 40 45.527 85.1897 -01 56 33.26 -1.9426
Saiph 05 47 45.389 86.9391 -09 40 10.58 -9.6696
Sirius 06 45 08.917 101.2872 -16 42 58.02 -16.7161