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### Detailed Solution:
To determine if an angle is coterminal with [tex]\(-60^\circ\)[/tex], we need to understand that coterminal angles differ by a multiple of [tex]\(360^\circ\)[/tex]. Two angles [tex]\(\theta_1\)[/tex] and [tex]\(\theta_2\)[/tex] are coterminal if:
[tex]\[ \theta_1 = \theta_2 + 360n \][/tex]
where [tex]\(n\)[/tex] is an integer. In simple terms, we will subtract [tex]\(-60^\circ\)[/tex] from each given angle and check if the result is a multiple of [tex]\(360^\circ\)[/tex]:
1. [tex]\(-1020^\circ\)[/tex]:
[tex]\[ -1020 - (-60) = -1020 + 60 = -960 \][/tex]
Check if [tex]\(-960\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{-960}{360} = -2.6667 \quad \text{(not an integer)} \][/tex]
2. [tex]\(-780^\circ\)[/tex]:
[tex]\[ -780 - (-60) = -780 + 60 = -720 \][/tex]
Check if [tex]\(-720\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{-720}{360} = -2 \quad \text{(an integer)} \][/tex]
3. [tex]\(-420^\circ\)[/tex]:
[tex]\[ -420 - (-60) = -420 + 60 = -360 \][/tex]
Check if [tex]\(-360\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{-360}{360} = -1 \quad \text{(an integer)} \][/tex]
4. [tex]\(60^\circ\)[/tex]:
[tex]\[ 60 - (-60) = 60 + 60 = 120 \][/tex]
Check if [tex]\(120\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{120}{360} = 0.3333 \quad \text{(not an integer)} \][/tex]
5. [tex]\(120^\circ\)[/tex]:
[tex]\[ 120 - (-60) = 120 + 60 = 180 \][/tex]
Check if [tex]\(180\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{180}{360} = 0.5 \quad \text{(not an integer)} \][/tex]
6. [tex]\(300^\circ\)[/tex]:
[tex]\[ 300 - (-60) = 300 + 60 = 360 \][/tex]
Check if [tex]\(360\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{360}{360} = 1 \quad \text{(an integer)} \][/tex]
7. [tex]\(420^\circ\)[/tex]:
[tex]\[ 420 - (-60) = 420 + 60 = 480 \][/tex]
Check if [tex]\(480\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{480}{360} = 1.3333 \quad \text{(not an integer)} \][/tex]
8. [tex]\(660^\circ\)[/tex]:
[tex]\[ 660 - (-60) = 660 + 60 = 720 \][/tex]
Check if [tex]\(720\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{720}{360} = 2 \quad \text{(an integer)} \][/tex]
Thus, the angles that are coterminal with [tex]\(-60^\circ\)[/tex] are:
[tex]\[ -780^\circ, -420^\circ, 300^\circ, 660^\circ \][/tex]
To determine if an angle is coterminal with [tex]\(-60^\circ\)[/tex], we need to understand that coterminal angles differ by a multiple of [tex]\(360^\circ\)[/tex]. Two angles [tex]\(\theta_1\)[/tex] and [tex]\(\theta_2\)[/tex] are coterminal if:
[tex]\[ \theta_1 = \theta_2 + 360n \][/tex]
where [tex]\(n\)[/tex] is an integer. In simple terms, we will subtract [tex]\(-60^\circ\)[/tex] from each given angle and check if the result is a multiple of [tex]\(360^\circ\)[/tex]:
1. [tex]\(-1020^\circ\)[/tex]:
[tex]\[ -1020 - (-60) = -1020 + 60 = -960 \][/tex]
Check if [tex]\(-960\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{-960}{360} = -2.6667 \quad \text{(not an integer)} \][/tex]
2. [tex]\(-780^\circ\)[/tex]:
[tex]\[ -780 - (-60) = -780 + 60 = -720 \][/tex]
Check if [tex]\(-720\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{-720}{360} = -2 \quad \text{(an integer)} \][/tex]
3. [tex]\(-420^\circ\)[/tex]:
[tex]\[ -420 - (-60) = -420 + 60 = -360 \][/tex]
Check if [tex]\(-360\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{-360}{360} = -1 \quad \text{(an integer)} \][/tex]
4. [tex]\(60^\circ\)[/tex]:
[tex]\[ 60 - (-60) = 60 + 60 = 120 \][/tex]
Check if [tex]\(120\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{120}{360} = 0.3333 \quad \text{(not an integer)} \][/tex]
5. [tex]\(120^\circ\)[/tex]:
[tex]\[ 120 - (-60) = 120 + 60 = 180 \][/tex]
Check if [tex]\(180\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{180}{360} = 0.5 \quad \text{(not an integer)} \][/tex]
6. [tex]\(300^\circ\)[/tex]:
[tex]\[ 300 - (-60) = 300 + 60 = 360 \][/tex]
Check if [tex]\(360\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{360}{360} = 1 \quad \text{(an integer)} \][/tex]
7. [tex]\(420^\circ\)[/tex]:
[tex]\[ 420 - (-60) = 420 + 60 = 480 \][/tex]
Check if [tex]\(480\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{480}{360} = 1.3333 \quad \text{(not an integer)} \][/tex]
8. [tex]\(660^\circ\)[/tex]:
[tex]\[ 660 - (-60) = 660 + 60 = 720 \][/tex]
Check if [tex]\(720\)[/tex] is a multiple of [tex]\(360\)[/tex]:
[tex]\[ \frac{720}{360} = 2 \quad \text{(an integer)} \][/tex]
Thus, the angles that are coterminal with [tex]\(-60^\circ\)[/tex] are:
[tex]\[ -780^\circ, -420^\circ, 300^\circ, 660^\circ \][/tex]
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