To determine which expression finds the measure of an angle that is coterminal with a [tex]\(126^\circ\)[/tex] angle, let's start by understanding what coterminal angles are. Coterminal angles are angles that share the same initial and terminal sides but may differ by full rotations. In degrees, a full rotation is [tex]\(360^\circ\)[/tex].
Therefore, an angle that is coterminal with [tex]\(126^\circ\)[/tex] can be found by adding or subtracting multiples of [tex]\(360^\circ\)[/tex]. Mathematically, this can be expressed as:
[tex]\[126^\circ + 360n^\circ\][/tex]
where [tex]\(n\)[/tex] is any integer (positive, negative, or zero).
Let's analyze the given options to determine which one matches this condition:
1. [tex]\(126^\circ + 275n^\circ\)[/tex]
2. [tex]\(126^\circ + 375n^\circ\)[/tex]
3. [tex]\(126^\circ + 450n^\circ\)[/tex]
4. [tex]\(126^\circ + 720n^\circ\)[/tex]
- Option 1: [tex]\(126^\circ + 275n^\circ\)[/tex]:
- Here, [tex]\(275^\circ\)[/tex] is not a multiple of [tex]\(360^\circ\)[/tex]. Since [tex]\(275\)[/tex] does not represent a full rotation, this expression does not properly find coterminal angles.
- Option 2: [tex]\(126^\circ + 375n^\circ\)[/tex]:
- Similarly, [tex]\(375^\circ\)[/tex] is not a multiple of [tex]\(360^\circ\)[/tex]. The difference between two angles would include additional rotations, which means this isn't valid for finding exactly coterminal angles.
- Option 3: [tex]\(126^\circ + 450n^\circ\)[/tex]:
- [tex]\(450^\circ\)[/tex] is also not a multiple of [tex]\(360^\circ\)[/tex]. Therefore, this expression represents rotations that go beyond the exact number of full circles needed to return to the same terminal side, and hence it does not work for coterminal angles.
- Option 4: [tex]\(126^\circ + 720n^\circ\)[/tex]:
- Here, [tex]\(720^\circ\)[/tex] is indeed a multiple of [tex]\(360^\circ\)[/tex]. In fact, [tex]\(720 = 2 \times 360\)[/tex]. This means each multiple of [tex]\(720^\circ\)[/tex] represents two full rotations, thereby ensuring that [tex]\(126^\circ\)[/tex] plus any multiple of [tex]\(720^\circ\)[/tex] will always be coterminal with [tex]\(126^\circ\)[/tex].
Hence, the expression that correctly finds the measure of an angle coterminal with a [tex]\(126^\circ\)[/tex] angle is:
[tex]\[126^\circ + (720n)^\circ\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{4}
\][/tex]
