Question

if 3cosx+4sinx=Asin(x+@),then find value of A and alpha??

Answer 1

3cosx + 4sinx = Asin(x + Ф)

Multiply and divide with $\bold{\sqrt{3^2+4^2}}$ = 5 in LHS
e.g., (3cosx + 4sinx)/5 = Asin(x + Ф)
⇒5(3/5.cosx + 4/5.sinx) = Asin(x + Ф)

Let cosФ= 3/5 then, sinФ = 4/5
Now, tanФ = sinФ/cosФ = 4/3
⇒ Ф = tan⁻¹(4/3) ----(1)

And 5(3/5.cosx + 4/5.sinx)= 5(sinФ.cosx + cosФ.sinx) = Asin(x + Ф)
⇒ 5sin(x + Ф) = Asin(x + Ф)
From equation (1),
⇒ 5sin(x + tan⁻¹(4/3)) = Asin(x + Ф)
Compare both sides,
A = 5 and Ф = tan⁻¹(4/3)