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You are given the equation y(t)=2sin4πt+5cos4πt, which models the position of the weight, with respect to time.
You need to find the amplitude of the oscillation, the angular frequency, and the initial conditions of the motion.
You will also be required to find the time(s) at which the weight is at a particular position.
To find this information, you need to convert the equation to the first form, y(t)=Asin(wt+Φ).
Question: Use the information above and the trigonometric identities to prove that Asin(wt+Φ)=c2sinwt+c1coswt .
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There is an absolutely standard method solving such problems.
The mathematicians, physicists and electrical engineers know it very well - they do it automatically.
See below and watch attentively each my step.
(1) You re-write the original equation step by step in this form
y(t) = 2*sin(4πt) + 5*cos(4πt) = . (1)
(2) Consider the coefficients = and = .
Notice that the coefficients and are positive and + = 1.
Therefore, there is an angle Φ in the first quadrant QI such that cos(Φ) = , sin(Φ) = .
Simply Φ = .
(3) Therefore, we can re-write (1) in the form
y(t) = = *(cos(Φ)*sin(4πt) + sin(Φ)*cos(4πt)) (2)
(4) Next, apply the formula for sine of the sum of arguments. You can continue writing the formula (2) in this way
y(t) = *sin(4πt + Φ).
(5) Now compare it with your formula y(t) = Asin(wt+Φ).
You see that the amplitude A = , w = 4π and the phase shift Φ = .
The solution is completed.
To get the numerical values, use your calculator.