University of The People Pre Calculus Problems Homework please see attached documents for specificsThere are two common forms of the equations used to model simple harmonic motion (SHM), which is the motion of springs, swings, tides, and many other periodic phenomena. These equations arE where:y(t) = distance of weight from equilibrium position = angular frequency (measured in radians per second)A = amplitude = phase (depends on initial conditions)c1 = Asinc2 = Acos Suppose you are an engineer trying to recreate an experiment involving a weight on the end of a spring. This simulation will give you an idea of what the experiment will look like. For more information, you can visit this simple harmonic motionwebsite. 8:47 PM Fri Apr 17

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Unit Activity: Trigonometric Functions

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Investigation of Simple Harmonic
Motion
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Hola
There are two common forms of the equations used to model simple harmonic
motion (SHM), which is the motion of springs, swings, tides, and many other periodic
phenomena. These equations are y(t) = A sin(wt+) and
y(t) = ca sin wt + ci cos wt, where:
y(t) = distance of weight from equilibrium position
w = angular frequency (measured in radians per second)
A = amplitude
0 = phase (depends on initial conditions)
(1 = Asing
C2 = Acoso
Suppose you are an engineer trying to recreate an experiment involving a weight on
the end of a spring. This simulation will give you an idea of what the experiment
will look like. For more information, you can visit this simple harmonic motion e
website.
You are given the equation y(t) = 2 sin 47t + 5 cos 4nt, 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+).
Part A
Use the information above and the trigonometric identities to prove that
A sin(wt+0) = C2 sin wt + ci cos wt.
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Unit Activity: Trigonometric Functions
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Part B

To rewrite y(t) = 2 sin 4At + 5 cos 47t in the form y(t) = A sin(wt+o),

you must first find the amplitude, A. Use the given values ci A sino and

C2 = A cosd, along with the Pythagorean identity, to solve for A.

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Part C

To rewrite y(t) = 2 sin 4nt + 5 cos 4nt in the form y(t) = A sin(wt+),

solve for 0.

Show your work and solution in the response box.

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Unit Activity: Trigonometric Functions

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Part D
Write y(t) = 2 sin 4t + 5 cos 4at in the form y(t) = A sin(wt+0) and
identify the amplitude, angular frequency, and the phase shift of the spring
motion.
Record your answers in the response box.
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Part E
The angle o represents the phase shift, determined by the initial conditions of
the experiment or the position of the weight at t = 0. If the weight is at its
maximum positive position (weight is above equilibrium) at t = 0, then 0 = 0. If
the weight is at its maximum negative position (spring is stretched and weight is
below equilibrium) at t = 0, then $ = 7. If the weight is traveling in the negative
direction and passing through equilibrium at t= 0, then $ = . In the response
box, describe the initial condition of our experiment; specifically, describe the
position of the weight and the direction in which it was traveling.
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Part F
Find the times (to the nearest hundredth of a second) that the weight is halfway
to its maximum negative position over the interval 0 < t < 0.5. Solve
algebraically, and show your work and final answer in the response box. Hint:
Use the amplitude to determine what y(t) must be when the weight is halfway to
its maximum negative position. Graph the equation and explain how it confirms
your solution(s).
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