A sine wave is described by the function
f(x) = Asin(Bx+C) + Df(x)=Asin(Bx+C)+D
where AA, BB, CC and DD are given constants.
We ask; how can we turn our function into this form? Well, notice how our function is of the form
f(x) = asinu(x)+bcosu(x)f(x)=asinu(x)+bcosu(x)
Where u(x)u(x) is another function in terms of xx. To make it easier to read, let u= u(x)u=u(x). Suppose there exists omega>0ω>0 and tauτ such that
asinu+bcosu=omegasin(u+tau)asinu+bcosu=ωsin(u+τ)
As there is no constant term in the formula for f(x)f(x) and the coefficient of uu is 11, we don't need to add additional constants.
omega(sinu+tau) = omegasinucostau+omegacosusintauω(sinu+τ)=ωsinucosτ+ωcosusinτ
color(red)asinu + color(blue)bcos u = color(red)(omegacostau)sinu + color(blue)(omegasintau)cosuasinu+bcosu=ωcosτsinu+ωsinτcosu
=> {(omegacostau=a),(omegasintau=b) :}
Square both relations and add them to reach the condition:
omega^2(sin^2tau+cos^2tau) = a^2+b^2=> omega=sqrt(a^2+b^2)
Dividing the second relation by the first yields
tan tau = b/a=> tau=arctan(b/a)
Hence
asinu+bcosu = sqrt(a^2+b^2)sin(u+arctan b"/"a)
f(x) = 7cos(1/3x)+sqrt19sin(1/3x)
=sqrt((sqrt19)^2+(7)^2)sin(1/3x + arctan 7"/"sqrt19)
=sqrt68sin(1/3x+arctan7"/"sqrt19)
Proving that f defines a sinusoid.
