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Law of Sines and Law of Cosines

Solving Triangles Without Right Angles

Standard trigonometry (SOH CAH TOA) is limited to right-angled triangles. But in the real world—from surveying land to navigating airplanes—triangles rarely have perfect 90-degree corners. These are called oblique triangles.

To solve these, mathematicians use two powerful generalizations: the Law of Sines and the Law of Cosines. These formulas relate the sides and angles of any triangle.

1. Labeling the Triangle

Before applying these laws, it is crucial to label the triangle correctly using standard convention:

[Image of oblique triangle labeled ABC]
  • Vertices (Angles): Capital letters A, B, and C.
  • Sides: Lowercase letters a, b, and c.
  • Opposite Pairs: Side 'a' must be opposite Angle 'A', side 'b' opposite Angle 'B', etc.

2. The Law of Sines

The Law of Sines is a ratio. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

[Image of Law of Sines formula]
a / sin(A) = b / sin(B) = c / sin(C)

When to Use It:

Use the Law of Sines when you have a "matching pair" (a known angle and its known opposite side) plus one other piece of information.

  • AAS (Angle-Angle-Side): You know two angles and a non-included side.
  • ASA (Angle-Side-Angle): You know two angles and the included side.
  • SSA (Side-Side-Angle): You know two sides and a non-included angle. (Note: This is the "Ambiguous Case" and may result in 0, 1, or 2 triangles).

3. The Law of Cosines

The Law of Cosines is essentially the Pythagorean Theorem adapted for non-right triangles. It relates all three sides to one angle.

[Image of Law of Cosines formula]
c² = a² + b² - 2ab cos(C)

You can rewrite this to find any side (e.g., a² = b² + c² - 2bc cos(A)).

When to Use It:

Use the Law of Cosines when you do not have a matching pair.

  • SAS (Side-Angle-Side): You know two sides and the angle trapped between them ("the sandwich").
  • SSS (Side-Side-Side): You know all three sides and need to find an angle.

4. Comparison Guide

Choosing the right law is half the battle. Here is a quick decision flow:

  • Do you have a right angle? -> Use SOH CAH TOA.
  • Do you have a side and its opposite angle? -> Use Law of Sines.
  • Do you have two sides and the included angle (SAS)? -> Use Law of Cosines.
  • Do you have all three sides (SSS)? -> Use Law of Cosines.

Conclusion

The Law of Sines and Law of Cosines are essential tools in advanced geometry. They allow us to calculate distances and angles in any triangular shape, making them vital for GPS technology, architecture, and physics.