The greatest & the least magnitude of the resultant of two forces of constant magnitudes are P & Q respectively. If the forces act at an angle `2 alpha`, then prove that the magnitude of their resultant is given by `sqrt(P^(2) cos^(2) alpha+Q^(2) sin^(2) alpha)`.

Explanation

Let two forces are A and B then
`A+B=PA-B=Q rArr A=(P+Q)/2, B=(P-Q)/2`
Resultant `k=sqrt(A^(2)+B^(2)+2AB cos theta)`
`=sqrt(((P+Q)/2)^(2)+((P-Q)/2)^(2)+2((P+Q)/2)((P-Q)/2) cos 2alpha)`
`=sqrt(P^(2)/2+Q^(2)/2+1/2(P^(2)-Q^(2))cos 2 alpha)`
`=sqrt(P^(2)/2(1+cos 2alpha)+Q^(2)/2(1-cos 2 alpha))`
`=sqrt(P^(2) cos^(2)alpha+Q^(2) sin^(2) alpha)`