Question 1169739: I don't know what the topic is
Background info:
You are given the equation y(t)=2sin(4πt) + 5cos(4πt), 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)=A sin(wt+Φ)
More background info:
y(t) = distance of weight from equilibrium position
w= angular frequency (measured in radians per second)
A = amplitude
Φ = phase (depends on initial conditions)
c1 = AsinΦ
c2 = AcosΦ
Question: To rewrite 2sin(4πt) + 5cos(4πt) in the form y(t)=A sin(wt+ Φ), you must first find the amplitude, A. Use the given values c1=AsinΦ and c2=AcosΦ, along with the Pythagorean identity, to solve for A.
Found 2 solutions by htmentor, ikleyn:
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! y(t)=2sin(4πt) + 5cos(4πt)
In order to rewrite the equation in the form y(t)=A sin(wt+ Φ), we use the
identity y(t) = AcosΦsin(wt) + AsinΦcos(wt)
Thus c1 = AcosΦ = 2, c2 = AsinΦ = 5
If we divide the 1st by the 2nd, we have:
tanΦ = 5/2 = 2.5 -> Φ = 1.1903
Therefore, A = 5/sinΦ = 5.3852
So the equation of motion is y(t) = 5.3852*sin(wt + 1.1903)
The angular frequency, w = 4π
The period, T = 2π/w = 1/2, and the initial condition is y(0) = 5/sinΦ*sin(Φ) = 5.
One can also use the Pythagorean identity, as the problem suggests, to get the
amplitude: c1^2 + c2^2 = 25 + 4 = A^2(sin^2Φ + cos^2Φ) -> A = sqrt(29)
Answer by ikleyn(52906)  (Show Source):
You can put this solution on YOUR website! .
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.
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