We now explore the relationship between A sin(wt) + B cos(t) and C sin(t + ). (a)

We now explore the relationship between A sin(wt) + B cos(ωt) and C sin(ωt + ϕ).
(a) By expanding sin(wt + ϕ) using the sum of the angles formula, show that the two expressions are equivalent if A = C cos ϕ and B = C sin ϕ.
(b) Consequently, show that A2 + B2 = C2 and that 4, then satisfies the equation tan ϕ = B/A.
(c) Generalize your result to state a proposition about A1 sin(ωt + ϕ1) A2 sin(ωt + ϕ2) + A3 sin(wt + ϕ3).
(d) Write an essay, in your own words, that expresses the importance of the identity between A sin(wt) + B cos(wt) and C sin(wt + ϕ). Be sure to note that |C| ≥ max(|A|. |B|) and that the identity holds only when you are forming a linear combination (adding andlor subtracting multiples of single powers) of sine and cosine of the same frequency.