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Trig Word Problem

A carousel can be modeled by the equation $y = 1.5 \cdot \sin \left(2 \cdot t + c\right)$ where the phase shift is adjusted based on the starting position. If the horse reached a maximum height after $\frac{\pi}{2}$ seconds, find the equation that models the horse’s position.

$\begin{equation*} y = a \cdot \sin \left( b \cdot t + c \right) + d \end{equation*}$

$\begin{equation*} y = 1.5 \cdot \sin \left( 2 \cdot t + c \right) \end{equation*}$

$\begin{equation*} a = 1.5 \end{equation*}$

$\begin{equation*} b = 2 \end{equation*}$

$\begin{equation*} c = \text{?} \end{equation*}$

$\begin{equation*} d = 0 \end{equation*}$

$\begin{equation*} \text{phase shift } = \frac{-c}{|b|} \end{equation*}$

$\begin{equation*} \frac{\pi}{2} = \frac{-\text{?}}{|2|} \end{equation*}$

$\begin{equation*} 2 \cdot \left( \frac{\pi}{2} \right) = \left( \frac{-\text{?}}{|2|} \right) \cdot 2 \end{equation*}$

$\begin{equation*} \frac{\pi}{-1} = \frac{-\text{?}}{-1} \end{equation*}$

$\begin{equation*} \text{? } = -\pi \end{equation*}$

$\begin{equation*} c = -\pi \end{equation*}$

$\begin{equation*} \boxed{y = 1.5 \cdot \sin \left(2 \cdot t - \pi\right)} \end{equation*}$