Proof of the Law of Cosines
The Law of Cosines states that for any triangle ABC, with sides a,b,c
For more see Law of Cosines.
- In the right triangle BCD, from the definition of cosine:
- Subtracting this from the side b, we see that
- In the triangle BCD, from the definition of sine:
- In the triangle ADB, applying the Pythagorean Theorem
- Substituting for BD and DA from (2) and (3)
- Multiplying out the parentheses:
- Rearranging the terms:
- Factoring out a2
- Looking at the terms in the parentheses above, recall that this is one of the trig identities, which states that
(See Pythagorean trig identites.)
So the terms in the parentheses can be removed since mutiplying a2 by one leaves it unchanged.
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