The height, [tex]d[/tex], in feet of a ball suspended from a spring as a function of time, [tex]t[/tex], in seconds can be modeled by the equation

[tex]\[ d = -2 \sin \left(\pi \left(t + \frac{1}{2}\right)\right) + 5 \][/tex]

The ball is released from its lowest point at [tex]t = 0[/tex] seconds. Using your knowledge of the general form of sine and cosine functions, which of the following equations can also model this situation?

A. [tex]d = -2 \cos (\pi t) + 5[/tex]

B. [tex]d = -2 \cos \left(\pi \left(t + \frac{1}{2}\right)\right) + 5[/tex]

C. [tex]d = 2 \cos (\pi t) + 5[/tex]

D. [tex]d = 2 \cos \left(\pi \left(t + \frac{1}{2}\right)\right) + 5[/tex]



Answer :

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]