what approximate radical as sqrt((r.^2)+(n.^2)-(2.*n.*r.*cos(x)))

Question

How to be simple radical as sqrt((r.^2)+(d.^2)-(2.*r.*d.*cos(x)))

 

 

Answer 1

 

√(r²+d²-2rdcos(x))=

r(1+(d²-2rdcos(x))/r²)^½≃(using binomial expansion to first degree):

r(1+(d²-2rdcos(x))/(2r²))=

r+(d²-2rdcos(x))/(2r)=

r+½d²/r-dcos(x).

The accuracy of the approximation improves the smaller d is in comparison to r.

The expression reduces further to r-dcos(x).

The original radical appears to be an application of the Cosine Rule to find the length of the third side of a triangle when the lengths of the other two sides and the included angle are known.