Question
90°-X is on which quadrant
X+360° is on which quadrant
X+360° is on which quadrant
Answers
GPT-4o mini
To determine the quadrants for the angles \(90^\circ - X\) and \(X + 360^\circ\), we need to analyze them based on the value of \(X\).
1. **For \(90^\circ - X\)**:
- If \(X\) is between \(0^\circ\) and \(90^\circ\), then \(90^\circ - X\) will be between \(90^\circ\) and \(0^\circ\), which means it lies in the **first quadrant**.
- If \(X = 90^\circ\), then \(90^\circ - X = 0^\circ\), which is on the boundary.
- If \(X\) is more than \(90^\circ\), then \(90^\circ - X\) will result in a negative angle, which corresponds to angles in the **fourth quadrant** when converted to a positive angle.
- If \(X = 180^\circ\), then \(90^\circ - X = -90^\circ\), which is again on the boundary of the fourth quadrant.
- For \(X > 180^\circ\), \(90^\circ - X\) continues to be negative and will remain in the **fourth quadrant**.
2. **For \(X + 360^\circ\)**:
- Adding \(360^\circ\) to any angle \(X\) does not change the terminal side of the angle. It simply makes it a full rotation plus the original angle \(X\).
- Therefore, the quadrant of \(X + 360^\circ\) is the same as the quadrant of \(X\).
- If \(X\) is between \(0^\circ\) and \(90^\circ\), it is in the **first quadrant**.
- If \(X\) is between \(90^\circ\) and \(180^\circ\), it is in the **second quadrant**.
- If \(X\) is between \(180^\circ\) and \(270^\circ\), it is in the **third quadrant**.
- If \(X\) is between \(270^\circ\) and \(360^\circ\), it is in the **fourth quadrant**.
In summary:
- \(90^\circ - X\) will lie in the first quadrant for \(0^\circ < X < 90^\circ\) and in the fourth quadrant for \(X > 90^\circ\).
- \(X + 360^\circ\) will have the same quadrant as the original angle \(X\).
1. **For \(90^\circ - X\)**:
- If \(X\) is between \(0^\circ\) and \(90^\circ\), then \(90^\circ - X\) will be between \(90^\circ\) and \(0^\circ\), which means it lies in the **first quadrant**.
- If \(X = 90^\circ\), then \(90^\circ - X = 0^\circ\), which is on the boundary.
- If \(X\) is more than \(90^\circ\), then \(90^\circ - X\) will result in a negative angle, which corresponds to angles in the **fourth quadrant** when converted to a positive angle.
- If \(X = 180^\circ\), then \(90^\circ - X = -90^\circ\), which is again on the boundary of the fourth quadrant.
- For \(X > 180^\circ\), \(90^\circ - X\) continues to be negative and will remain in the **fourth quadrant**.
2. **For \(X + 360^\circ\)**:
- Adding \(360^\circ\) to any angle \(X\) does not change the terminal side of the angle. It simply makes it a full rotation plus the original angle \(X\).
- Therefore, the quadrant of \(X + 360^\circ\) is the same as the quadrant of \(X\).
- If \(X\) is between \(0^\circ\) and \(90^\circ\), it is in the **first quadrant**.
- If \(X\) is between \(90^\circ\) and \(180^\circ\), it is in the **second quadrant**.
- If \(X\) is between \(180^\circ\) and \(270^\circ\), it is in the **third quadrant**.
- If \(X\) is between \(270^\circ\) and \(360^\circ\), it is in the **fourth quadrant**.
In summary:
- \(90^\circ - X\) will lie in the first quadrant for \(0^\circ < X < 90^\circ\) and in the fourth quadrant for \(X > 90^\circ\).
- \(X + 360^\circ\) will have the same quadrant as the original angle \(X\).