There is a family of sinusoidal functions of the form:
[tex]y = 4 + 6\cdot \cos [(2+4\cdot i)\cdot x], \forall \,i \in \mathbb{Z}[/tex]
Where [tex]i[/tex] is associated to the angular frequency of a function within the family.
In this case, the sinusoidal function reaches the upper bound when [tex]x = 0[/tex] and the middle value when [tex]x = 0.25\pi[/tex]. We should find all coefficients contained in the following sinusoidal model:
[tex]y = (y_{max}-y_{mid})\cdot \cos A\cdot x + B[/tex] (1)
Where:
- [tex]x[/tex] - Independent value.
- [tex]y[/tex] - Dependent value.
- [tex]y_{mid}[/tex] - Midpoint value.
- [tex]y_{max}[/tex] - Upper bound value.
- [tex]A[/tex] - Period coefficient.
- [tex]B[/tex] - Translation coefficient.
If we know that [tex]y_{mid} = 4[/tex], [tex]y_{max} = 10[/tex], [tex](x_{1},y_{1}) = (0, 10)[/tex] and [tex](x_{2}, y_{2}) = \left(\frac{\pi}{4}, 4 \right)[/tex], then we have the following system of equations:
[tex]6+ B = 10[/tex] (2)
[tex]6\cdot \cos \frac{\pi\cdot A}{4} + B = 4[/tex] (3)
By (2):
[tex]B = 4[/tex]
By (3):
[tex]6\cdot \cos \frac{\pi\cdot A}{4} = 0[/tex]
[tex]\cos \frac{\pi\cdot A}{4} = 0[/tex]
[tex]\frac{\pi\cdot A}{4} = \cos ^{-1} 0[/tex]
[tex]A = \frac{4}{\pi}\cdot \cos^{-1}0[/tex]
[tex]A = \frac{4}{\pi}\cdot \left(\frac{\pi}{2}+\pi\cdot i \right)[/tex], [tex]\forall \,i\in\mathbb{Z}[/tex]
[tex]A = 2 +4\cdot i[/tex], [tex]\forall \,i \in \mathbb{Z}[/tex]
There is a family of sinusoidal functions of the form:
[tex]y = 4 + 6\cdot \cos [(2+4\cdot i)\cdot x], \forall \,i \in \mathbb{Z}[/tex] (4)
Where [tex]i[/tex] is associated to the angular frequency of a function within the family.
We kindly invite to check this question on sinusoidal functions: https://brainly.com/question/12060967
