Intro

Tl;dr, the law of sines is:

\frac{\sin(\alpha)}{a} = \frac{\sin(\beta)}{b} = \frac{\sin(\gamma)}{c}

Under convention:

Angle \alpha is opposite side a
Angle \beta is opposite side b
Angle \gamma is opposite side c
Any triangle that is not a right triangle is called an oblique triangle. There are two types of oblique triangles:
- Acute triangles: This is an oblique triangle where all three interior angles are less than 90\degree or \dfrac{\pi}{2} radians.
- Obtuse Triangle: This is an oblique triangle where one of the interior angles is greater than 90\degree.

Different types of oblique triangles

ASA Triangle: (Angle Side Angle) - We know the measurements of two angles and the side between them
AAS: We know the measurements of two angles and a side that is not between the known angles.
SSA: We know the measurements of two sides and an angle that is not between the known sides. These triangles can be solved by adding a line that goes from one vertex to intersect perpendicular to the opposite side, forming two right triangles (h).

We know that \sin\alpha = \dfrac{h}{b} and \sin\beta = \dfrac{h}{a}. We can sole both equations for h to get:

h = b\sin\alpha
h = a\sin\beta Setting both equations equal to each other gives us: b\sin\alpha = a\sin\beta

Multiply both sides by \dfrac{1}{ab} gives yields \dfrac{\sin\alpha}{a} = \dfrac{\sin\beta}{b}

Side side angle triangles may be solved to have one possible solution, two possible solutions, or no possible solutions.