Solving the right triangle

Dependencies: Definition of sin / cos / tan, Pythagorean theorem, Triangle interior angles sum to 180°.

Statement

A right triangle $\triangle ABC$ ($\angle C = 90^\circ$) has 5 unknowns in total: two acute angles $\angle A$, $\angle B$ and three sides $a = BC$ (opposite $\angle A$), $b = CA$ (adjacent to $\angle A$), $c = AB$ (the hypotenuse). These 5 quantities are subject to two structural constraints:

After removing the two constraints, 3 free parameters remain; subtracting one more for the "overall similarity-scaling" degree of freedom leaves an effective degree of freedom equal to 2.

Conclusion. Specifying any 2 non-trivial elements — provided the 2 are not both acute angles — uniquely determines the remaining three quantities via Definition of sin / cos / tan and Pythagorean theorem. The three concrete branches:

• (a) One acute angle + one side: the other acute angle $= 90^\circ - \theta$; the remaining two sides come directly from $\sin\theta$, $\cos\theta$, $\tan\theta$.
• (b) Two sides: use Pythagorean theorem to find the third side, then read off the two acute angles via inverse trig functions.
• (c) Two acute angles: insufficient degrees of freedom (locks only the shape, not the size), unsolvable.