Sum of sinusoids at the same frequency but with different amplitudes and phases

Question

Three sinusoids at the same frequency are superimposed:

Cos[3.1 - t] + 5 Cos[π/4 + t] + 2 Sin[0.3 + t]

How can the resulting oscillation be stated in the following forms?

a Cos[t] + b Sin[t] and A Cos[t + α]

I have tried many things; even asked chatgpt and I just cannot figure it out. Any help is greatly appreciated.

Answer 1

2 Sin[t + 0.3] + 5 Cos[t + π/4] + Cos[t - 3.1] // TrigExpand // Simplify

3.12744 Cos[t] - 1.58328 Sin[t]

a = 3.12744; b = -1.58328;
φ1 = 0; φ2 = -π/2;

amp = Sqrt[a^2 + b^2 + 2*a*b*Cos[φ2 - φ1]];
ϕ = ArcTan[a*Cos[φ1] + b*Cos[φ2], a*Sin[φ1] + b*Sin[φ2]] // N;
amp*Cos[ϕ + t]

3.50538 Cos[0.468639 + t]

Answer 2

e1 = 2 Sin[t + 0.3];
e2 = 5 Cos[t + π/4];
e3 = Cos[t - 3.1];
expr = e1 + e2 + e3 
FullSimplify[expr] 

3.12744 Cos[t] - 1.58328 Sin[t]

res = FullSimplify[expr] /. 
  a_. Cos[t] + b_. Sin[t] :> a Sqrt[1 + b^2/a^2] Cos[t - ArcTan[b/a]]

3.50537 Cos[0.468639 + t]

---

Plot[{e1, e2, e3, expr, res}, {t, 0, 6 π}, 
 PlotStyle -> {Automatic, Automatic, Automatic
   , {Opacity[0.6, Red], Thickness[0.02]}, {Thick, Blue}}
 , PlotLegends -> "Expressions"
 ]

---

How to arrive at the replacement formula:

{TrigExpand[A Cos[t - ϕ]], a Cos[t] + b Sin[t]}

{A Cos[t] Cos[ϕ] + A Sin[t] Sin[ϕ], a Cos[t] + b Sin[t]}

Comparing the two and solving:

sol = First[
    Assuming[{a, b, A, ϕ} ∈ Reals, 
     Solve[{A Cos[ϕ] == a , 
       A Sin[ϕ] == b }, {A, ϕ}]]] /. C[1] -> 1;

A Cos[t - ϕ] /. sol

Answer 3

Or you can hit it with a hammer.

$Version

(* "13.3.1 for Mac OS X ARM (64-bit) (July 24, 2023)" *)

Clear["Global`*"]

f[t_] = Cos[3.1 - t] + 5 Cos[π/4 + t] + 2 Sin[0.3 + t] // Rationalize;

data = Table[{t, f[t]}, {t, 0, 10, 1/10}];

(param1 = NonlinearModelFit[data,
     a Cos[t] + b Sin[t], {a, b}, t,
     WorkingPrecision -> 15]["BestFitParameters"]) // N

(* {a -> 3.12744, b -> -1.58328} *)

f[t] == a Cos[t] + b Sin[t] /. param1 // Simplify

(* True *)

(param2 = NonlinearModelFit[data,
     A Cos[t + α], {A, α}, t,
     WorkingPrecision -> 20]["BestFitParameters"]) // N

(* {A -> 3.50537, α -> 0.468639} *)

f[t] == A Cos[t + α] /. param2 // Simplify

(* True *)