what are the amplitude and period of the sinusoid y=6cosx+7sinx?

Answer 1

We could write 6cosx + 7sinx as a single
sine function of the form asin(x + A)

asin(x+A) = asinxcosA + acosxsinA = 7sinx + 6cosx
let x=0
---> asinA = 6 or sinA = 6/a
let x=π/2
---> acosA = 7 or cosA = 7/a
but sin^2 A + cos^2 B = 1
36/a^2 + 49/a^2 = 1
a^2 = 85
a =√85

so the amplitude is √85
and the period is 2π, (A is simply a phase shift)