Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all.

Question

Solve each triangle that results. a=12, b=14, A=51°

Answer 1

When we are given the angle opposite the shorter side, we can have, if any triangles, likely two (unless we have a right triangle).

From the sin law, B = sin^-1(14 * sin(51°)/12) = 65.0491908146464 or 180°-65.0491908146464° = 114.950809185354°

Rounding to the nearest degree, 65° and 115°, respectively, C will be 180° - 51° - 65° = 64° or

180° - 51° - 115° = 14°

Our triangle has angles 51-65-64 or 51-115-14

Two triangles are possible.

Answer 2

law of cosines says:

c^2 = a^2 + b^2 - 2ab cos C

= 12^2 + 14^2 - 2(12)(14)cos 51

= 144 + 196 - 336cos51

17 < c < 18 per the calculator