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:

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.

Answer:

• (A) \(\cos(\pi - \theta) = -\cos\theta\)
• (B) \(\sin(\pi + \theta) = -\sin\theta\)
• (C) \(\sin(\pi + \theta) = \sin\theta\) (Correct answer)
• (D) \(\sin(\pi - \theta) = \sin\theta\)