The standard ratios (SOH CAH TOA) are excellent tools, but they have a major limitation: they only work on triangles with a 90° angle. However, the real world is full of Oblique Triangles (triangles with no right angle, either acute or obtuse).
To solve these, we use two powerful generalizations of trigonometry: the Law of Sines and the Law of Cosines.
1. Labeling Oblique Triangles
Unlike right triangles where we use "Hypotenuse," oblique triangles are labeled by their vertex letters.
- Angles: Capital letters (A, B, C).
- Sides: Lowercase letters (a, b, c), where side 'a' is opposite angle 'A'.
2. The Law of Sines
The Law of Sines describes the relationship between the length of a side and the sine of its opposite angle. It works best when you have a "matching pair" (a side and its opposite angle).
When to use it:
- AAS or ASA: You know two angles and a side.
- SSA: You know two sides and a non-included angle (Careful! This is the "Ambiguous Case" and can sometimes result in 0, 1, or 2 solutions).
3. The Law of Cosines
The Law of Cosines is the "big brother" of the Pythagorean Theorem. It relates the three sides of a triangle to the cosine of one of its angles. It is perfect for finding a side when you know the "included" angle (the angle trapped between two known sides).
When to use it:
- SAS: You know two sides and the angle between them.
- SSS: You know all three sides and want to find an angle.
4. Area of a Non-Right Triangle
Normally, finding the area requires ½ × base × height. But finding the vertical "height" in an oblique triangle can be tedious. Trigonometry offers a shortcut using two sides and the included angle.
This means if you know two sides (a and b) and the angle (C) sitting between them, you can find the area instantly without knowing the height.
Conclusion
Non-Right Triangle Trig expands our ability to measure the world. From calculating flight paths to triangulating GPS signals, the Laws of Sines and Cosines allow us to solve any triangle, regardless of its shape.