The equation given is [tex]\( d = -2 \sin \left( \pi \left( t + \frac{1}{2} \right) \right) + 5 \)[/tex].
To find an equivalent equation involving a cosine function, we can use the trigonometric identity:
[tex]\[
\sin \left( x + \frac{\pi}{2} \right) = \cos(x)
\][/tex]
First, let's manipulate the argument of the sine function:
[tex]\[
\sin \left( \pi \left( t + \frac{1}{2} \right) \right) = \sin \left( \pi t + \frac{\pi}{2} \right)
\][/tex]
Now, using the identity, we replace [tex]\(\sin \left( \pi t + \frac{\pi}{2} \right)\)[/tex] with [tex]\(\cos (\pi t)\)[/tex]:
[tex]\[
\sin \left( \pi t + \frac{\pi}{2} \right) = \cos (\pi t)
\][/tex]
Therefore, the original equation [tex]\( d = -2 \sin \left( \pi \left( t + \frac{1}{2} \right) \right) + 5 \)[/tex] can be rewritten as:
[tex]\[
d = -2 \cos (\pi t) + 5
\][/tex]
So, let's compare this result with the provided options:
1. [tex]\( d = -2 \cos (\pi t) + 5 \)[/tex]
2. [tex]\( d = -2 \cos \left( \pi \left( t + \frac{1}{2} \right) \right) + 5 \)[/tex]
3. [tex]\( d = 2 \cos (\pi t) + 5 \)[/tex]
4. [tex]\( d = 2 \cos \left( \pi \left( t + \frac{1}{2} \right) \right) + 5 \)[/tex]
Among these options, the equation that correctly models the situation and is equivalent to the given sine function is:
[tex]\[
d = -2 \cos (\pi t) + 5
\][/tex]
Therefore, the correct answer is the first option:
[tex]\[
\boxed{1}
\][/tex]
