To find the measures of all angles coterminal with [tex]\(-220^\circ\)[/tex], we need to find an angle that lies between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex] which is coterminal with [tex]\(-220^\circ\)[/tex].
Step-by-Step Solution:
1. Understanding Coterminal Angles:
- Two angles are coterminal if, when drawn in standard position, their terminal sides coincide. This happens when they differ by an integer multiple of [tex]\(360^\circ\)[/tex].
2. Finding the Reference Angle:
- To find a positive coterminal angle with [tex]\(-220^\circ\)[/tex], we can add [tex]\(360^\circ\)[/tex] (a full circle) until we get a positive angle.
3. Calculation:
- Add [tex]\(360^\circ\)[/tex] to [tex]\(-220^\circ\)[/tex]:
[tex]\[
-220^\circ + 360^\circ = 140^\circ
\][/tex]
- Therefore, [tex]\(140^\circ\)[/tex] is a positive angle that is coterminal with [tex]\(-220^\circ\)[/tex].
4. General Form of Coterminal Angles:
- To express all angles coterminal with [tex]\(-220^\circ\)[/tex], we can use the formula:
[tex]\[
\theta + 360n
\][/tex]
- Substituting [tex]\(\theta = 140^\circ\)[/tex]:
[tex]\[
140^\circ + 360n
\][/tex]
- Here, [tex]\(n\)[/tex] can be any integer (positive, negative, or zero) to cover all possible coterminal angles.
5. Final Answer:
- Therefore, the measure of all angles coterminal with [tex]\(-220^\circ\)[/tex] is represented as:
[tex]\[
140 + 360n, \text{ for any integer } n
\][/tex]
So, the correct answer from the choices given is:
[tex]\[
\boxed{140+360n, \text{ for any integer } n}
\][/tex]
