To determine the measures of all angles coterminal with a [tex]\(-220^\circ\)[/tex] angle, we need to understand the concept of coterminal angles. Coterminal angles share the same initial and terminal sides but may differ by multiples of [tex]\(360^\circ\)[/tex] since [tex]\(360^\circ\)[/tex] represents a full rotation.
Here's a step-by-step solution to find the answer:
1. Find a positive coterminal angle for [tex]\(-220^\circ\)[/tex]:
We can find a positive coterminal angle by adding [tex]\(360^\circ\)[/tex] (one full rotation) to [tex]\(-220^\circ\)[/tex]:
[tex]\[
-220^\circ + 360^\circ = 140^\circ
\][/tex]
So, [tex]\(140^\circ\)[/tex] is a positive coterminal angle for [tex]\(-220^\circ\)[/tex].
2. Generalize the coterminal angles:
To find all coterminal angles, we add integer multiples of [tex]\(360^\circ\)[/tex] to the [tex]\(140^\circ\)[/tex]:
[tex]\[
\theta = 140^\circ + 360n
\][/tex]
where [tex]\(n\)[/tex] is an integer ([tex]\(n\)[/tex] can be positive, negative, or zero).
3. Identify the correct representation:
The correct representation should include all integer multiples of [tex]\(360^\circ\)[/tex].
Therefore, the measure of all angles coterminal with [tex]\(-220^\circ\)[/tex] is correctly represented by:
[tex]\[
140 + 360n \text{, for any integer } n
\][/tex]
So the correct answer is:
[tex]\[
\boxed{140 + 360n \text{, for any integer } n}
\][/tex]
