To determine the sign of \(\sin\left(\frac{5\pi}{4}\right)\) and the quadrant in which it lies, let's analyze the angle:
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Identify the angle: The angle \(\frac{5\pi}{4}\) radians can be converted to degrees for better visualization: \[ \frac{5\pi}{4} \times \frac{180}{\pi} = 225^\circ. \]
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Determine the quadrant: Angles are measured counterclockwise from the positive x-axis. The angle \(225^\circ\) lies in the third quadrant (where angles from \(180^\circ\) to \(270^\circ\) are located).
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Determine the sign of \(\sin\left(\frac{5\pi}{4}\right)\): In the third quadrant, the sine function is negative. Therefore: \[ \sin\left(\frac{5\pi}{4}\right) < 0. \]
In summary:
- The sine of \(\frac{5\pi}{4}\) is negative, i.e., \(\sin\left(\frac{5\pi}{4}\right) < 0\).
- The angle \(\frac{5\pi}{4}\) is in the third quadrant.