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education/math/MATH1060 (trig)/Law of Sines.md
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# 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 $90\degree$.
- **Obtuse Triangle**: This is an oblique triangle where one of the interior angles is greater than $90\degree$.
## Different types of oblique triangles
1. **ASA Triangle**: (Angle Side Angle) - We know the measurements of two angles and the side between them
2. **AAS**: We know the measurements of two angles and a side that is not between the known angles.
3. **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 ($h$).
## Solving for the law of sines
We know that $\sin\alpha = \dfrac{h}{b}$