To solve the problem of determining the relationship between angle [tex]\(x\)[/tex] and angle [tex]\(y\)[/tex] when [tex]\(x\)[/tex] is coterminal with [tex]\(y\)[/tex] and the measure of angle [tex]\(x\)[/tex] is greater than angle [tex]\(y\)[/tex], let's first understand what it means for two angles to be coterminal.
1. Definition of Coterminal Angles:
- Two angles are said to be coterminal if they share the same terminal side. This means that they can differ by a full revolution (or multiple full revolutions) around the circle.
- A full revolution in degrees is 360°.
2. Mathematical Expression:
- Given an angle [tex]\(y\)[/tex], any angle [tex]\(x\)[/tex] that is coterminal with [tex]\(y\)[/tex] can be expressed as:
[tex]\[
x = y + 360k
\][/tex]
where [tex]\(k\)[/tex] is an integer. This equation means that angle [tex]\(x\)[/tex] is obtained by adding or subtracting multiples of 360° to/from angle [tex]\(y\)[/tex].
3. Considering [tex]\(x > y\)[/tex]:
- Since it is given that the measure of angle [tex]\(x\)[/tex] is greater than the measure of angle [tex]\(y\)[/tex], we need to add multiples of 360° to [tex]\(y\)[/tex].
- Therefore, [tex]\(k\)[/tex] must be a positive integer so that when we add [tex]\(360k\)[/tex] to [tex]\(y\)[/tex], the resultant angle [tex]\(x\)[/tex] is greater than [tex]\(y\)[/tex].
4. Possible Statements Analysis:
- [tex]\(x = y - 180n\)[/tex] for any positive integer [tex]\(n\)[/tex]: This would not necessarily produce a coterminal angle because subtracting 180° does not yield a full circle equivalent.
- [tex]\(x = y - 380n\)[/tex] for any integer [tex]\(n\)[/tex]: This incorrect because 380° is not a full revolution and subtracting it does not ensure coterminal angles.
- [tex]\(x = y + 360n\)[/tex] for any positive integer [tex]\(n\)[/tex]: This is correct because adding 360° (or multiples of it) maintains the coterminal property and makes [tex]\(x\)[/tex] greater than [tex]\(y\)[/tex].
- [tex]\(x = y + 180n\)[/tex] for any integer [tex]\(n\)[/tex]: Adding 180° can sometimes produce an angle not necessarily coterminal because 180° is not a full revolution.
Hence, the correct statement regarding the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[
x = y + 360n, \text{for any positive integer } n.
\][/tex]
