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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 f)+5[/tex]

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

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

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

Sagot :

GinnyAnswer
To determine which equation can also model the given situation, we need to transform the given equation involving the sine function into an equivalent equation using the cosine function. The given equation is:

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

Let's follow these steps to find an equivalent form:

1. Understanding the sine to cosine transformation:
We know that the sine function can be expressed in terms of the cosine function using the following identity:
[tex]\[ \sin(x) = \cos\left(\frac{\pi}{2} - x\right) \][/tex]

2. Apply the identity to the given equation:
So, we can rewrite the argument [tex]\(\pi \left(t + \frac{1}{2}\right)\)[/tex] inside the sine function using the identity:
[tex]\[ \sin\left(\pi \left(t + \frac{1}{2}\right)\right) = \cos\left(\frac{\pi}{2} - \pi \left(t + \frac{1}{2}\right)\right) \][/tex]

3. Simplifying the argument inside the cosine function:
Simplify the expression inside the cosine function step-by-step:
[tex]\[ \cos\left(\frac{\pi}{2} - \pi \left(t + \frac{1}{2}\right)\right) = \cos\left(\frac{\pi}{2} - \pi t - \frac{\pi}{2}\right) \][/tex]
[tex]\[ \cos\left(\frac{\pi}{2} - \pi t - \frac{\pi}{2}\right) = \cos\left(-\pi t\right) \][/tex]
Since cosine is an even function, meaning [tex]\(\cos(-x) = \cos(x)\)[/tex], we get:
[tex]\[ \cos\left(-\pi t\right) = \cos\left(\pi t\right) \][/tex]

4. Substitute back into the original equation:
Now we can replace the original sine term with the cosine term:
[tex]\[ d = -2 \sin \left(\pi \left(t + \frac{1}{2}\right)\right) + 5 = -2 \cos\left(\pi t\right) + 5 \][/tex]

After this derivation, we can see that:

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

Now let's compare this with the provided options:

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

The correct option matching our derived equation is:

[tex]\[ \boxed{-2 \cos(\pi t) + 5} \][/tex]

Thus, the answer corresponding to the provided multiple-choice question is:

[tex]\[ \boxed{2} \][/tex]
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