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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 [tex]y(t)=A \sin (\omega t+\phi)[/tex] and [tex]y(t)=c_2 \sin \omega t+c_1 \cos \omega t[/tex], where:

- [tex]y(t)[/tex] = distance of weight from equilibrium position
- [tex]\omega[/tex] = angular frequency (measured in radians per second)
- [tex]A[/tex] = amplitude
- [tex]\phi[/tex] = phase (depends on initial conditions)
- [tex]c_1 = A \sin \phi[/tex]
- [tex]c_2 = A \cos \phi[/tex]

Suppose you are an engineer trying to recreate an experiment involving a weight on the end of a spring. You are given the equation [tex]y(t)=2 \sin 4 \pi t+5 \cos 4 \pi t[/tex], 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, [tex]y(t)=A \sin (\omega t+\phi)[/tex].

Part A
Use the information above and the trigonometric identities to prove that [tex]A \sin (\omega t+\phi)=c_2 \sin \omega t+c_1 \cos \omega t[/tex].



Answer :

GinnyAnswer
Let's start by proving that the equation [tex]\( A \sin(\omega t + \phi) = c_2 \sin \omega t + c_1 \cos \omega t \)[/tex].

First, recall the trigonometric identity for the sine of a sum of angles:
[tex]\[ \sin(\omega t + \phi) = \sin \omega t \cos \phi + \cos \omega t \sin \phi \][/tex]

Now, let's multiply both sides of this identity by the amplitude [tex]\( A \)[/tex]:
[tex]\[ A \sin(\omega t + \phi) = A (\sin \omega t \cos \phi + \cos \omega t \sin \phi) \][/tex]
[tex]\[ = A \cos \phi \sin \omega t + A \sin \phi \cos \omega t \][/tex]

We can identify the terms [tex]\( A \cos \phi \)[/tex] and [tex]\( A \sin \phi \)[/tex]. Let's define these as follows:
[tex]\[ c_2 = A \cos \phi \][/tex]
[tex]\[ c_1 = A \sin \phi \][/tex]

Substituting these values back into the equation, we get:
[tex]\[ A \sin(\omega t + \phi) = c_2 \sin \omega t + c_1 \cos \omega t \][/tex]

Thus, we have shown that:
[tex]\[ A \sin(\omega t + \phi) = c_2 \sin \omega t + c_1 \cos \omega t \][/tex]

Now, let's use this form to find the amplitude [tex]\( A \)[/tex], angular frequency [tex]\( \omega \)[/tex], and phase [tex]\( \phi \)[/tex] for the given equation [tex]\( y(t) = 2 \sin(4 \pi t) + 5 \cos(4 \pi t) \)[/tex].

Given:
[tex]\[ c_2 = 2 \][/tex]
[tex]\[ c_1 = 5 \][/tex]

### 1. Amplitude ([tex]\( A \)[/tex])
The amplitude [tex]\( A \)[/tex] is given by:
[tex]\[ A = \sqrt{c_1^2 + c_2^2} \][/tex]
Substitute the given values [tex]\( c_1 = 5 \)[/tex] and [tex]\( c_2 = 2 \)[/tex]:
[tex]\[ A = \sqrt{5^2 + 2^2} \][/tex]
[tex]\[ A = \sqrt{25 + 4} \][/tex]
[tex]\[ A = \sqrt{29} \][/tex]
[tex]\[ A \approx 5.385 \][/tex]

### 2. Angular Frequency ([tex]\( \omega \)[/tex])
The given equation has [tex]\( \omega t = 4 \pi t \)[/tex], so:
[tex]\[ \omega = 4 \pi \][/tex]
[tex]\[ \omega \approx 12.566 \][/tex]

### 3. Phase ([tex]\( \phi \)[/tex])
The phase [tex]\( \phi \)[/tex] can be found using the relationship:
[tex]\[ \phi = \arctan \left( \frac{c_1}{c_2} \right) \][/tex]
Substitute the given values [tex]\( c_1 = 5 \)[/tex] and [tex]\( c_2 = 2 \)[/tex]:
[tex]\[ \phi = \arctan \left( \frac{5}{2} \right) \][/tex]
[tex]\[ \phi \approx 1.190 \][/tex]

Therefore, the original equation [tex]\( y(t) = 2 \sin(4 \pi t) + 5 \cos(4 \pi t) \)[/tex] can be rewritten in the form [tex]\( y(t) = A \sin(\omega t + \phi) \)[/tex] as:
[tex]\[ y(t) \approx 5.385 \sin(12.566 t + 1.190) \][/tex]

We successfully identified the amplitude, angular frequency, and phase as:
- Amplitude [tex]\( A \approx 5.385 \)[/tex]
- Angular frequency [tex]\( \omega \approx 12.566 \)[/tex]
- Phase [tex]\( \phi \approx 1.190 \)[/tex]

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