To determine the measures of all angles coterminal with a [tex]\(400^{\circ}\)[/tex] angle, let's start by understanding the concept of coterminal angles. Coterminal angles are angles that share the same initial and terminal sides, but they can have different measures. These can be found by adding or subtracting multiples of [tex]\(360^{\circ}\)[/tex] to the given angle, because a full rotation in a circle is [tex]\(360^{\circ}\)[/tex].
Given an angle [tex]\(\theta\)[/tex], angles coterminal with [tex]\(\theta\)[/tex] can be expressed as [tex]\(\theta + 360n\)[/tex] where [tex]\(n\)[/tex] is any integer (positive, negative, or zero).
For the given angle [tex]\(400^{\circ}\)[/tex], we need to find it in its simplest coterminal form within one full circle rotation (i.e., between [tex]\(0^{\circ}\)[/tex] and [tex]\(360^{\circ}\)[/tex]). This can be done by subtracting [tex]\(360^{\circ}\)[/tex] from [tex]\(400^{\circ}\)[/tex] to get:
[tex]\[ 400^{\circ} - 360^{\circ} = 40^{\circ} \][/tex]
So, [tex]\(40^{\circ}\)[/tex] is a coterminal angle with [tex]\(400^{\circ}\)[/tex].
To represent all possible coterminal angles with [tex]\(400^{\circ}\)[/tex], we add multiples of [tex]\(360^{\circ}\)[/tex] to this result:
[tex]\[ 40^{\circ} + 360n \][/tex]
where [tex]\(n\)[/tex] is any integer.
Thus, the measures of all angles coterminal with a [tex]\(400^{\circ}\)[/tex] angle can be represented as [tex]\(40 + 360n\)[/tex] for any integer [tex]\(n\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{40 + 360n, \text{ for any integer } n} \][/tex]
