The equation that represents the sinusoidal function is [tex]x(t) = -8 + 6.5 \cdot \sin \left[\left(\frac{2}{3} \pm \frac{4\cdot i}{3}\right)\cdot t + \left(\frac{2\pi}{3} \pm \frac{7\pi \cdot i}{3} \right)\right][/tex], [tex]i\in \mathbb{Z}[/tex].
In this question we must find an appropriate model for a periodic function based on the information from statement. Sinusoidal functions are the most typical functions which intersects a midline ([tex]x_{mid}[/tex]) and has both a maximum ([tex]x_{max}[/tex]) and a minimum ([tex]x_{min}[/tex]).
Sinusoidal functions have in most cases the following form:
[tex]x(t) = x_{mid} + \left(\frac{x_{max}-x_{min}}{2} \right)\cdot \sin (\omega \cdot t + \phi)[/tex] (1)
Where:
If we know that [tex]x_{min} = -14.5[/tex], [tex]x_{mid} = -8[/tex], [tex]x_{max} = -1.5[/tex], [tex](t, x) = (-\pi, -8)[/tex] and [tex](t, x) = \left(\frac{\pi}{4}, -1.5 \right)[/tex], then the sinusoidal function is:
[tex]-8 +6.5\cdot \sin (-\pi\cdot \omega + \phi) = -8[/tex] (2)
[tex]-8+6.5\cdot \sin\left(\frac{\pi}{4}\cdot \omega + \phi \right) = -1.5[/tex] (3)
The resulting system is:
[tex]\sin (-\pi\cdot \omega + \phi) = 0[/tex] (2b)
[tex]\sin \left(\frac{\pi}{4}\cdot \omega + \phi \right) = 1[/tex] (3b)
By applying inverse trigonometric functions we have that:
[tex]-\pi\cdot \omega + \phi = 0 \pm \pi\cdot i[/tex], [tex]i \in \mathbb{Z}[/tex] (2c)
[tex]\frac{\pi}{4}\cdot \omega + \phi = \frac{\pi}{2} + 2\pi\cdot i[/tex], [tex]i \in \mathbb{Z}[/tex] (3c)
And we proceed to solve this system:
[tex]\pm \pi\cdot i + \pi\cdot \omega = \frac{\pi}{2} \pm 2\pi\cdot i -\frac{\pi}{4}\cdot \omega[/tex]
[tex]\frac{3\pi}{4}\cdot \omega = \frac{\pi}{2}\pm \pi\cdot i[/tex]
[tex]\omega = \frac{2}{3} \pm \frac{4\cdot i}{3}[/tex], [tex]i\in \mathbb{Z}[/tex] [tex]\blacksquare[/tex]
By (2c):
[tex]-\pi\cdot \left(\frac{2}{3} \pm \frac{4\cdot i}{3}\right) + \phi =\pm \pi\cdot i[/tex]
[tex]-\frac{2\pi}{3} \mp \frac{4\pi\cdot i}{3} + \phi = \pm \pi\cdot i[/tex]
[tex]\phi = \frac{2\pi}{3} \pm \frac{7\pi\cdot i}{3}, i\in \mathbb{Z}[/tex] [tex]\blacksquare[/tex]
The equation that represents the sinusoidal function is [tex]x(t) = -8 + 6.5 \cdot \sin \left[\left(\frac{2}{3} \pm \frac{4\cdot i}{3}\right)\cdot t + \left(\frac{2\pi}{3} \pm \frac{7\pi \cdot i}{3} \right)\right][/tex], [tex]i\in \mathbb{Z}[/tex]. [tex]\blacksquare[/tex]
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