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Sagot :
To determine which of the given angles are coterminal with [tex]\(-50^\circ\)[/tex], we need to understand what coterminal angles are. Angles are coterminal if they differ by multiples of [tex]\(360^\circ\)[/tex]. This means for an angle [tex]\(\theta\)[/tex], another angle [tex]\(\theta' \)[/tex] will be coterminal if [tex]\( \theta' - \theta \)[/tex] is a multiple of [tex]\(360^\circ\)[/tex].
Given the angle [tex]\(-50^\circ\)[/tex], we're looking for angles that satisfy:
[tex]\[ (\theta' + 50) = 360n \][/tex]
where [tex]\(n\)[/tex] is an integer.
Checking each angle:
1. [tex]\(-770^\circ\)[/tex] :
[tex]\[ -770 + 50 = -720 \][/tex]
[tex]\[ -720 \div 360 = -2 \][/tex]
So, [tex]\(-770^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
2. [tex]\(-530^\circ\)[/tex] :
[tex]\[ -530 + 50 = -480 \][/tex]
[tex]\[ -480 \div 360 = -4/3 \quad (\text{not an integer}) \][/tex]
So, [tex]\(-530^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
3. [tex]\(-410^\circ\)[/tex] :
[tex]\[ -410 + 50 = -360 \][/tex]
[tex]\[ -360 \div 360 = -1 \][/tex]
So, [tex]\(-410^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
4. [tex]\(50^\circ\)[/tex] :
[tex]\[ 50 + 50 = 100 \][/tex]
[tex]\[ 100 \div 360 = 5/18 \quad (\text{not an integer}) \][/tex]
So, [tex]\(50^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
5. [tex]\(310^\circ\)[/tex] :
[tex]\[ 310 + 50 = 360 \][/tex]
[tex]\[ 360 \div 360 = 1 \][/tex]
So, [tex]\(310^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
6. [tex]\(360^\circ\)[/tex] :
[tex]\[ 360 + 50 = 410 \][/tex]
[tex]\[ 410 \div 360 = 1.138 \quad (\text{not an integer}) \][/tex]
So, [tex]\(360^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
7. [tex]\(410^\circ\)[/tex] :
[tex]\[ 410 + 50 = 460 \][/tex]
[tex]\[ 460 \div 360 = 1.278 \quad (\text{not an integer}) \][/tex]
So, [tex]\(410^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
8. [tex]\(670^\circ\)[/tex] :
[tex]\[ 670 + 50 = 720 \][/tex]
[tex]\[ 720 \div 360 = 2 \][/tex]
So, [tex]\(670^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
The angles that are coterminal with [tex]\(-50^\circ\)[/tex] are:
[tex]\[ -770^\circ, -410^\circ, 310^\circ, 670^\circ \][/tex]
Given the angle [tex]\(-50^\circ\)[/tex], we're looking for angles that satisfy:
[tex]\[ (\theta' + 50) = 360n \][/tex]
where [tex]\(n\)[/tex] is an integer.
Checking each angle:
1. [tex]\(-770^\circ\)[/tex] :
[tex]\[ -770 + 50 = -720 \][/tex]
[tex]\[ -720 \div 360 = -2 \][/tex]
So, [tex]\(-770^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
2. [tex]\(-530^\circ\)[/tex] :
[tex]\[ -530 + 50 = -480 \][/tex]
[tex]\[ -480 \div 360 = -4/3 \quad (\text{not an integer}) \][/tex]
So, [tex]\(-530^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
3. [tex]\(-410^\circ\)[/tex] :
[tex]\[ -410 + 50 = -360 \][/tex]
[tex]\[ -360 \div 360 = -1 \][/tex]
So, [tex]\(-410^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
4. [tex]\(50^\circ\)[/tex] :
[tex]\[ 50 + 50 = 100 \][/tex]
[tex]\[ 100 \div 360 = 5/18 \quad (\text{not an integer}) \][/tex]
So, [tex]\(50^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
5. [tex]\(310^\circ\)[/tex] :
[tex]\[ 310 + 50 = 360 \][/tex]
[tex]\[ 360 \div 360 = 1 \][/tex]
So, [tex]\(310^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
6. [tex]\(360^\circ\)[/tex] :
[tex]\[ 360 + 50 = 410 \][/tex]
[tex]\[ 410 \div 360 = 1.138 \quad (\text{not an integer}) \][/tex]
So, [tex]\(360^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
7. [tex]\(410^\circ\)[/tex] :
[tex]\[ 410 + 50 = 460 \][/tex]
[tex]\[ 460 \div 360 = 1.278 \quad (\text{not an integer}) \][/tex]
So, [tex]\(410^\circ\)[/tex] is not coterminal with [tex]\(-50^\circ\)[/tex].
8. [tex]\(670^\circ\)[/tex] :
[tex]\[ 670 + 50 = 720 \][/tex]
[tex]\[ 720 \div 360 = 2 \][/tex]
So, [tex]\(670^\circ\)[/tex] is coterminal with [tex]\(-50^\circ\)[/tex].
The angles that are coterminal with [tex]\(-50^\circ\)[/tex] are:
[tex]\[ -770^\circ, -410^\circ, 310^\circ, 670^\circ \][/tex]
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