Question
determine the sign of sin (5pie/4) and the quadranr in which it lies
Answers
GPT-4o mini
To determine the sign of \(\sin\left(\frac{5\pi}{4}\right)\) and the quadrant in which it lies, let's analyze the angle:
1. **Identify the angle**: The angle \(\frac{5\pi}{4}\) radians can be converted to degrees for better visualization:
\[
\frac{5\pi}{4} \times \frac{180}{\pi} = 225^\circ.
\]
2. **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).
3. **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.
1. **Identify the angle**: The angle \(\frac{5\pi}{4}\) radians can be converted to degrees for better visualization:
\[
\frac{5\pi}{4} \times \frac{180}{\pi} = 225^\circ.
\]
2. **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).
3. **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.