Worked example

A triangle has a = 7, b = 5 and included angle C = 60°. Find the area and then angle A via the sine rule

Answer

Area = 35√3/4; and the same area written with another base gives a sin C = c sin A

Area = ½ a b sin C
Two sides and the included angle. · Triangle area in two pairings
= ½ · 7 · 5 · (√3/2) = 35√3/4
sin 60° = √3/2.
½ a b sin C = ½ b c sin A
Same area, other pairing — this is the sine rule in disguise. · Triangle area in two pairings
a sin C = c sin A → sin A = a sin C / c
Cancel ½b; needs side c from the cosine rule first.
Note the constraint
a, b, c, A, B, C are not free numbers: A + B + C = 180° and the sides track the sines. Our sampler generates real triangles for exactly this reason.

½ a b sin C = ½ b c sin A

Triangle area in two pairings — Any two-side-plus-included-angle area computation in a triangle. Written equal like this, it is the sine rule in disguise: the area does not care which pair of sides you use.

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.