To determine which angle is coterminal with [tex]\(110^\circ\)[/tex], we need to find angles that, when added or subtracted by integer multiples of [tex]\(360^\circ\)[/tex] (a full rotation), produce the same terminal side as [tex]\(110^\circ\)[/tex].
The general formula for coterminal angles is:
[tex]\[ \theta_{\text{coterminal}} = \theta + 360k \][/tex]
where [tex]\(\theta\)[/tex] is the original angle and [tex]\(k\)[/tex] is any integer.
Let's evaluate each option to see if it is coterminal with [tex]\(110^\circ\)[/tex]:
### Option A: [tex]\(460^\circ\)[/tex]
[tex]\[ 460^\circ - 110^\circ = 350^\circ \][/tex]
[tex]\[ 350^\circ \div 360^\circ = 0.9722\ \text{(not an integer)} \][/tex]
This is not a multiple of [tex]\(360^\circ\)[/tex], hence [tex]\(460^\circ\)[/tex] is not coterminal with [tex]\(110^\circ\)[/tex].
### Option B: [tex]\(-110^\circ\)[/tex]
[tex]\[ -110^\circ - 110^\circ = -220^\circ \][/tex]
[tex]\[ -220^\circ \div 360^\circ = -0.6111\ \text{(not an integer)} \][/tex]
This is not a multiple of [tex]\(360^\circ\)[/tex], so [tex]\(-110^\circ\)[/tex] is not coterminal with [tex]\(110^\circ\)[/tex].
### Option C: [tex]\(470^\circ\)[/tex]
[tex]\[ 470^\circ - 110^\circ = 360^\circ \][/tex]
[tex]\[ 360^\circ \div 360^\circ = 1 \][/tex]
Since [tex]\(1\)[/tex] is an integer, [tex]\(470^\circ\)[/tex] is a coterminal angle with [tex]\(110^\circ\)[/tex].
### Option D: [tex]\(290^\circ\)[/tex]
[tex]\[ 290^\circ - 110^\circ = 180^\circ \][/tex]
[tex]\[ 180^\circ \div 360^\circ = 0.5\ \text{(not an integer)} \][/tex]
This is not a multiple of [tex]\(360^\circ\)[/tex], therefore [tex]\(290^\circ\)[/tex] is not coterminal with [tex]\(110^\circ\)[/tex].
Thus, the angle that is coterminal with [tex]\(110^\circ\)[/tex] is:
C. An angle measuring [tex]\(470^\circ\)[/tex]
