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.
