To determine the value and sign of \( \sin\left(\frac{5\pi}{4}\right) \), we need to identify the quadrant in which this angle lies.
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Finding the angle in degrees:
We convert \( \frac{5\pi}{4} \) to degrees:
\[ \frac{5\pi}{4} \times \frac{180}{\pi} = 5 \times 45 = 225^\circ \] -
Identifying the quadrant:
The angle \( 225^\circ \) lies in the third quadrant (between \( 180^\circ \) and \( 270^\circ \)). -
Determining the sign of sine:
In the third quadrant, the sine function is negative.
Therefore, we conclude that:
- \( \sin\left(\frac{5\pi}{4}\right) \) is negative.
- It lies in Quadrant III.
The correct response is:
sin(5π4) is negative and lies in Quadrant III.