To evaluate Marco's equation, we can first calculate \( \cos\left(-\frac{\pi}{2}\right) \) and \( \cos\left(\frac{3\pi}{2}\right) \).
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Calculate \( \cos\left(-\frac{\pi}{2}\right) \):
- The cosine function is even: \( \cos(-x) = \cos(x) \).
- Therefore, \( \cos\left(-\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \).
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Calculate \( \cos\left(\frac{3\pi}{2}\right) \):
- The angle \( \frac{3\pi}{2} \) is located on the negative y-axis, where the cosine value is also 0.
- Therefore, \( \cos\left(\frac{3\pi}{2}\right) = 0 \).
Since both sides of Marco's equation are equal (0 = 0), the equation is true.
Now, we analyze the reasoning:
- The statement "It is true, because the cosine function is odd" is incorrect; the cosine function is even.
- The statement "It is false, because the cosine function is even" is incorrect because the equation is actually true.
- The statement "It is true, because the cosine function has a period of \( 2\pi \)" is also not the relevant reasoning for the equivalency in this specific case.
- The statement "It is false, because the cosine function has a period of π" is not applicable since the correct reasoning is about the values at the specified angles.
The conclusion is:
The best response would be: "It is true, because the cosine function is even." This option highlights that both angles, when evaluated, yield the same cosine value based on the property of the even function. However, the phrasing "It is true, because the cosine function is odd" is inaccurate, as cosine is not odd.
Thus, the statement regarding the cosine function being even is the closest to correct logic, even if it doesn't explicitly appear among the response options.
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