Question

acosA-bsinA = c ,prove that asinA+bcos=+√a^+b^-c^

Answer 1

Given a cosA-b sinA=c
or, (a cosA-b sinA)^2=c^2

or, a^2 cos^2A -2ab cosAsinA + b^2 sin^2A=c^2

or, a^2 (1-sin^2A)+b^2 (1-cos^2A)=c^2+2ab cosAsinA

or, a^2-a^2 sin^2A +b^2-b^2 cos^2A= c^2+2ab sinAcosA

or, -a^2 sin^2A-b^2 cos^2A- 2ab sinAcosA= -a^2-b^2+c^2

or, a^2 sin^2A + b^2 cos^2A + 2ab sinAcosA = a^2+ b^2-c^2

or, ( a sinA + b cosA)^2=a^2+b^2-c^2

or, a sinA +b cosA= plus minus root over a^2+b^2-c^2. PROVED.