Question Number 101713 by I want to learn more last updated on 04/Jul/20

Let   a  and  b  be positive numbers satisfying   a^2   +  b^2   =  5,  If    a cos(θ)  −  b sin(θ)  =  1,     find    a sin(θ)  +  b(cosθ)

Answered by mr W last updated on 04/Jul/20

let  a=(√5) sin α  b=(√5) cos α  a sin θ+b cos θ=t  sin α cos θ−cos α sin θ=(1/(√5))   ...(i)  sin α sin θ+cos α cos θ=(t/(√5))  ...(ii)  (i)^2 +(ii)^2   cos^2  θ+sin^2  θ=(1/5)+(t^2 /5)=1  ⇒t=±2

Commented byI want to learn more last updated on 04/Jul/20

Thanks sir

Answered by Dwaipayan Shikari last updated on 04/Jul/20

a^2 +b^2 =1+x^2   x=±2

Commented byDwaipayan Shikari last updated on 04/Jul/20

Thanking you

Commented bymr W last updated on 04/Jul/20

good sir!

Commented byI want to learn more last updated on 04/Jul/20

Thanks sir

Answered by john santu last updated on 04/Jul/20

(1)a^2 cos ^2 θ−2abcos θsin θ+b^2 sin ^2  θ=1  (2)a^2 sin ^2 θ+2abcos θsin θ+b^2 cos ^2 θ=j^2   ________________________ +  5 = 1 +j^2  ⇒ j = ± 2   (JS ⊛)

Commented byI want to learn more last updated on 04/Jul/20

Thanks sir