Question

$3 \cos(theta) + 4 sin(theta) = a sin(theta + alpha )$

Answer 1

3cosθ+4sinθ = a sin(θ+α)

you can convert this to sin(a+b) form

sin(a+b) = sin(a)cos(b)+cos(a)sin(b)

multiply and divide by 5 on left hand side for the given equation

5( 3 ∕ 5 × cos(θ) + 4 ∕ 5 × sin(θ) ) = a sin(θ+α)

now consider a triangle with 3,4,5 as its sides

sinβ = 3 ∕ 5 ; β = sine inverse of 3 /5

cosβ = 4 / 5

5(sinβcosθ+cosβsinθ) = a sin(θ+α)

5(sin(β+θ)) = a sin(θ+α)

a = 5(sin(β+θ))/sin(θ+α)

Here if β = α then the value of a=5