#cosinerule

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[logicxonomy]

Cosine Formula Proof (Trigonometry Tricks)

Let, In a △ABC, the length of sides: BC=a, CA=b, and AB=c. BD is perpendicular drawn from vertex B to side AC, here h is the height of the triangle.

Then, According to Cosine Formula:

a²=b²+c²–2bc×Cos(A)

b²=c²+a²–2ac×Cos(B)

c²=a²+b²–2ab×Cos(C)

Here is the proof of this formula.

Apply Pythagoras Theorem in △BDC

a²=h²+DC²

R.H.S.

=h²+(AC-AD)²

=h²+AC²+AD²–2.AC.AD

=(h²+AD²)+b²–2.b.AD

=c²+b²–2.b.AD

Note: h²+AD²=c² (Pythagoras Theorem)

Note: In △ADB

Cos(A)=AD/AB=AD/c

AD=c.Cos(A)

So, a²=b²+c²–2.bc.Cos(A)

Similarly, we can prove other combinations also.

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