Question

کدام رابطه درست نیست؟

\\(\cos(\pi - \theta) = -\cos\theta\\)
\\(\sin(\pi + \theta) = -\sin\theta\\)
\\(\sin(\pi + \theta) = \sin\theta\\)
\\(\sin(\pi - \theta) = \sin\theta\\)

Explanation:

<pre_analysis>
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</pre_analysis>

<reasoning>

Analyze the first relation

\[
\cos(\pi - \theta) = -\cos\theta
\]
This is a correct trigonometric identity.

Analyze the second and third relations

\[
\sin(\pi + \theta) = -\sin\theta
\]
This is correct because \(\pi + \theta\) lies in the third quadrant where sine is negative.
\[
\sin(\pi + \theta) = \sin\theta
\]
This is incorrect because it contradicts the correct identity above.

Analyze the fourth relation

\[
\sin(\pi - \theta) = \sin\theta
\]
This is correct because \(\pi - \theta\) lies in the second quadrant where sine is positive.
</reasoning>

<answer>
<mcq-option>(A) \(\cos(\pi - \theta) = -\cos\theta\)</mcq-option>
<mcq-option>(B) \(\sin(\pi + \theta) = -\sin\theta\)</mcq-option>
<mcq-correct>(C) \(\sin(\pi + \theta) = \sin\theta\)</mcq-correct>
<mcq-option>(D) \(\sin(\pi - \theta) = \sin\theta\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Identities"
]
}
</post_analysis>

Answer:

<pre_analysis>
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],
"new_concepts": [],
"current_concepts": [
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}
</pre_analysis>

<reasoning>

Analyze the first relation

\[
\cos(\pi - \theta) = -\cos\theta
\]
This is a correct trigonometric identity.

Analyze the second and third relations

\[
\sin(\pi + \theta) = -\sin\theta
\]
This is correct because \(\pi + \theta\) lies in the third quadrant where sine is negative.
\[
\sin(\pi + \theta) = \sin\theta
\]
This is incorrect because it contradicts the correct identity above.

Analyze the fourth relation

\[
\sin(\pi - \theta) = \sin\theta
\]
This is correct because \(\pi - \theta\) lies in the second quadrant where sine is positive.
</reasoning>

<answer>
<mcq-option>(A) \(\cos(\pi - \theta) = -\cos\theta\)</mcq-option>
<mcq-option>(B) \(\sin(\pi + \theta) = -\sin\theta\)</mcq-option>
<mcq-correct>(C) \(\sin(\pi + \theta) = \sin\theta\)</mcq-correct>
<mcq-option>(D) \(\sin(\pi - \theta) = \sin\theta\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Identities"
]
}
</post_analysis>