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Angle [tex]\( x \)[/tex] is coterminal with angle [tex]\( y \)[/tex]. If the measure of angle [tex]\( x \)[/tex] is greater than the measure of angle [tex]\( y \)[/tex], which statement is true regarding the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]?

A. [tex]\( x = y - 180n \)[/tex], for any positive integer [tex]\( n \)[/tex]
B. [tex]\( x = y - 360n \)[/tex], for any integer [tex]\( n \)[/tex]
C. [tex]\( x = y + 360n \)[/tex], for any positive integer [tex]\( n \)[/tex]
D. [tex]\( x = y + 180n \)[/tex], for any integer [tex]\( n \)[/tex]

Sagot :

GinnyAnswer
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]
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