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Which of the following represents the measures of all angles coterminal with a [tex]\(-220^{\circ}\)[/tex] angle?

A. [tex]\(360 + 140n\)[/tex], for any whole number [tex]\(n\)[/tex]
B. [tex]\(360 + 140n\)[/tex], for any integer [tex]\(n\)[/tex]
C. [tex]\(140 + 360n\)[/tex], for any whole number [tex]\(n\)[/tex]
D. [tex]\(140 + 360n\)[/tex], for any integer [tex]\(n\)[/tex]


Sagot :

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]