Q: Prove that if cos(x) is an integer then so is sin(x).
A: The statement "if cos(x) is an integer, then so is sin(x)" is not true in general. To understand why, let’s consider the properties of sine and cosine. [...]
It had trouble when the problem domain is different from the usual one the same keywords typically fall into.
You:
Prove that if cos(x) is an integer then so is sin(x)
ChatGPT:
To prove this statement, let's first establish some basic facts about the sine and cosine functions:
1. *Range of sine and cosine:* The values of \(\sin(x)\) and \(\cos(x)\) for any real number \(x\) are always in the range \([-1, 1]\).
2. *Pythagorean Identity:* For any angle \(x\), \(\sin^2(x) + \cos^2(x) = 1\).
Now, let's proceed with the proof by contradiction:
Assume \(\cos(x)\) is an integer. Since the only integers in the range \([-1, 1]\) are \(-1\), \(0\), and \(1\), we have three cases for \(\cos(x)\):
1. If \(\cos(x) = -1\), then \(\sin^2(x) = 1 - \cos^2(x) = 1 - (-1)^2 = 0\). Hence, \(\sin(x) = 0\), which is an integer.
2. If \(\cos(x) = 0\), then \(\sin^2(x) = 1 - \cos^2(x) = 1 - 0^2 = 1\). Hence, \(\sin(x) = \pm 1\), which are integers.
3. If \(\cos(x) = 1\), then \(\sin^2(x) = 1 - \cos^2(x) = 1 - 1^2 = 0\). Hence, \(\sin(x) = 0\), which is an integer.
In each case, if \(\cos(x)\) is an integer, \(\sin(x)\) is also an integer. Therefore, it's proven that if \(\cos(x)\) is an integer, then so is \(\sin(x)\).
Q: Prove that if cos(x) is an integer then so is sin(x).
A: The statement "if cos(x) is an integer, then so is sin(x)" is not true in general. To understand why, let’s consider the properties of sine and cosine. [...]
It had trouble when the problem domain is different from the usual one the same keywords typically fall into.