To determine the sign of \(\sin\left(\frac{5\pi}{4}\right)\) and the quadrant in which it lies, we can follow these steps:
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Identify the angle: \(\frac{5\pi}{4}\) radians is equivalent to \(225^\circ\).
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Determine the quadrant: Angles between \(180^\circ\) and \(270^\circ\) (or between \(\pi\) and \(\frac{3\pi}{2}\) radians) lie in the third quadrant. Therefore, \(\frac{5\pi}{4}\) is in the third quadrant.
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Determine the sign of \(\sin\) in the third quadrant: In the third quadrant, the sine function is negative.
Thus, \(\sin\left(\frac{5\pi}{4}\right)\) is negative, and it lies in the third quadrant.
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