Deriving Law of Sines
and Law of Cosines
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Use Pythagorean Theorem
in a right triangle.

 

Our friend, the Pythagorean Theorem, comes to our aid when solving for sides and/or angles in a right triangle.  We have now seen that there are formulas (the Law of Sines and the Law of Cosines) that will help us find sides and/or angles when we are not working in a right triangle.

But where did these formulas come from?

Let's see how these formulas were derived:

Law of Sines


Triangle ABC at the right does not contain a right angle.  A perpendicular is dropped from vertex B.  It can now be observed that:

 

Now, drop a perpendicular from vertex A.  It can be observed that: 

Law of Sines: 
True for ALL triangles!



 

 

 

 

Law of Cosines


Triangle ABC at the right does not contain a right angle.  A perpendicular is dropped from vertex B.  It can now be observed that:

 



Using the Pythagorean Theorem in triangle CBD, we have:  .
Substituting for h and r we have:


 

This same process could be used to produce other lettered statements of this law.

Law of Cosines: 
A generalization of the Pythagorean Theorem.  If angle C were a right angle, the cosine of angle C would be zero and the Pythagorean Theorem would result.