To determine the quadrants for the angles \(90^\circ - X\) and \(X + 360^\circ\), we need to analyze them based on the value of \(X\).
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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.
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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 \(X\).