A sine wave is described by the function
#f(x) = Asin(Bx+C) + D#
where #A#, #B#, #C# and #D# 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)#
Where #u(x)# is another function in terms of #x#. To make it easier to read, let #u= u(x)#. Suppose there exists #omega>0# and #tau# such that
#asinu+bcosu=omegasin(u+tau)#
As there is no constant term in the formula for #f(x)# and the coefficient of #u# is #1#, we don't need to add additional constants.
#omega(sinu+tau) = omegasinucostau+omegacosusintau#
#color(red)asinu + color(blue)bcos u = color(red)(omegacostau)sinu + color(blue)(omegasintau)cosu#
#=> {(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.
