# Projection Law

Projection Law:

Projection law states that in any triangle: $b \cos C + c \cos B = a \\ \\ a \cos C + c \cos A = b \\ \\ a \cos B + b \cos A = c$

Where , A , B , C are the three angled of the triangle and a , b , c are the corresponding opposite side of the angles.

Projection law or the formula of projection law express the algebraic sum of the projection of any two side in term of the third side.

Proof of Projection law:

To prove the projection law we shall take the help of sine law which states that: $a = 2R \sin A , b = 2R \sin B , c = 2R \sin C$

Thus ,  using the above formula for sine law we can easily deduce the formula for projection law as: $b \cos C + c \cos B = 2R ( \sin B \cos C + \cos B \sin C )$

Now using the trigonometric addition formulas we can replace $( \sin B \cos C + \cos B \sin C )$ by $\sin ( B + C )$

And again in a triangle A+B+C = 180, so: $\sin (B + C ) = \sin (180 - A) = \sin A$

Now we can re write the original formula as: $b \cos C + c \cos B : \\ \\ = 2R ( \sin B \cos C + \cos B \sin C ) \\ \\ = 2R \sin A \\ \\ = a$

And we can also deduce the second and third formula for projection law similarly.

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