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Select all that are measures of angles that are coterminal with a [tex]\(-50^{\circ}\)[/tex] angle.

A. [tex]\(-770^{\circ}\)[/tex]
B. [tex]\(-530^{\circ}\)[/tex]
C. [tex]\(-410^{\circ}\)[/tex]
D. [tex]\(50^{\circ}\)[/tex]
E. [tex]\(310^{\circ}\)[/tex]
F. [tex]\(360^{\circ}\)[/tex]
G. [tex]\(410^{\circ}\)[/tex]
H. [tex]\(670^{\circ}\)[/tex]

Sagot :

GinnyAnswer
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]
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