Question

HELPPPP!!!

Answer 1

Answer:

  A = √29

Step-by-step explanation:

The short of it is that ...

  A² = 2² + 5² = 29

  A = √29

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Amplitude

If you expand the second form using the sum-of-angles formula, you get ...

  Asin(ωt +φ) = Asin(ωt)cos(φ) +Acos(ωt)sin(φ)

Comparing this to the first form, you find ...

  c₂ = 2 = Acos(φ)

  c₁ = 5 = Asin(φ)

The Pythagorean identity can be invoked to simplify the sum of squares:

  (Asin(φ))² + (Acos(φ))² = A²(sin(φ)² +cos(φ)²) = A²·1 = A²

In terms of c₁ and c₂, this is ...

  (c₁)² +(c₂)² = A²

  A = √((c₁)² +(c₂)²) . . . . . . . formula for amplitude

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Phase Shift

We know that tan(φ) = sin(φ)/cos(φ) = (Asin(φ))/(Acos(φ)) = 5/2, so ...

  φ = arctan(c₁/c₂) . . . . . . . formula for phase shift*

  φ = arctan(5/2) ≈ 1.19029 radians

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* remember that c₁ is the coefficient of the cosine term, and c₂ is the coefficient of the sine term.