Question Number 100131 by bemath last updated on 25/Jun/20

If 3sin x+4cos x=5 then sin x=?

Commented bybobhans last updated on 25/Jun/20

let tan ((x/2)) = t , ⇒sin x = ((2t)/(1+t^2 )) , cos x=((1−t^2 )/(1+t^2 ))  ⇔((6t+4−4t^2 )/(1+t^2 )) = 5 ⇒−4t^2 +6t+4=5t^2 +5  9t^2 −6t+1=0 , (3t−1)^2 =0  t=(1/3) ⇒sin x=((2/3)/(1+(1/9))) = (6/(10)) = (3/5)

Commented bybemath last updated on 25/Jun/20

thank you all

Commented byDwaipayan Shikari last updated on 25/Jun/20

(3sinx+4cosx)^2 =25  9sin^2 x+24sinxcosx+16cos^2 x=25(sin^2 x+cos^2 x)  16sin^2 x−24sinxcosx+9cos^2 x=0  (4sinx−3cosx)^2 =0  or  cotx=(4/3)  3+4cotx=(5/(sinx))   or   ((sinx)/5)=(3/(25))   so  sinx=(3/5)

Answered by Rasheed.Sindhi last updated on 25/Jun/20

If 3sin x+4cos x=5 then sin x=?          4cos x=5−3sin x          16cos^2  x=(5−3sin x)^2    16(1−sin^2 x)=25−30sin x+9sin^2 x     25sin^2 x−30sin x+9=0   sin x=((30±(√(900−900)))/(50)) =(3/5)

Answered by mr W last updated on 25/Jun/20

cos x cos α+sin x sin α=1  with sin α=(3/5), cos α=(4/5)  cos (x−α)=1  x−α=2kπ  x=2kπ+α  sin x=sin α=(3/5)