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Sagot :
To determine which negative angle is equivalent to an [tex]$85^{\circ}$[/tex] angle, we need to find angles that, when adjusted by adding or subtracting multiples of [tex]$360^{\circ}$[/tex], result in an angle of [tex]$85^{\circ}$[/tex].
Here’s a step-by-step solution:
1. Understanding Angles and their Equivalents:
- Any angle θ is equivalent to angles θ ± [tex]$360^{\circ}$[/tex], θ ± [tex]$720^{\circ}$[/tex], and so on.
- This means to find the negative equivalent of [tex]$85^{\circ}$[/tex], we can subtract multiples of [tex]$360^{\circ}$[/tex] from [tex]$85^{\circ}$[/tex] until we get a negative angle.
2. Calculation:
- Subtract [tex]$360^{\circ}$[/tex] from [tex]$85^{\circ}$[/tex]:
[tex]\[ 85^{\circ} - 360^{\circ} = -275^{\circ} \][/tex]
- This tells us that [tex]$-275^{\circ}$[/tex] is one equivalent negative angle for [tex]$85^{\circ}$[/tex].
- We can also subtract another multiple of [tex]$360^{\circ}$[/tex]:
[tex]\[ 85^{\circ} - 720^{\circ} = -635^{\circ} \text{ (and so on)} \][/tex]
3. Verify Each Choice:
- Choice A: [tex]$-85^{\circ}$[/tex]
- Check [tex]\( -85^{\circ} = 85^{\circ} - 360^{\circ} \)[/tex]:
[tex]\[ 85^{\circ} - 360^{\circ} \neq -85^{\circ} \][/tex]
- [tex]\( -85^{\circ} \)[/tex] is not equivalent to [tex]\( 85^{\circ} \)[/tex].
- Choice B: [tex]$-265^{\circ}$[/tex]
- Check [tex]\( -265^{\circ} \)[/tex]:
[tex]\[ -265^{\circ} \text { as equivalent to another angle } = 85^{\circ} - 360^{\circ} * x \][/tex]
- Adding [tex]\( 360^{\circ} \)[/tex]:
[tex]\[ -265^{\circ} + 360^{\circ} = 95^{\circ} \][/tex]
- This is not equivalent to [tex]\( 85^{\circ} \)[/tex].
- Choice C: [tex]$-95^{\circ}$[/tex]
- Check [tex]\( -95^{\circ} \)[/tex]:
[tex]\[ before calculation, it seems irrelevant. - \[ 85^{\circ} - 360^{\circ} \neq -95^{\circ} \][/tex]
- Choice D: [tex]$-275^{\circ}$[/tex]
- Check [tex]\( -275^{\circ} \)[/tex]:
[tex]\[ 85^{\circ} - 360^{\circ} \][/tex]
- Correct, one full rotation leads to [tex]\( 85^{\circ} - 360^{\circ} = -275^{\circ} \)[/tex].
Therefore, among the given options, the negative angle equivalent to [tex]\(85^{\circ}\)[/tex] is:
D. An angle measuring [tex]$-275^{\circ}$[/tex]
Here’s a step-by-step solution:
1. Understanding Angles and their Equivalents:
- Any angle θ is equivalent to angles θ ± [tex]$360^{\circ}$[/tex], θ ± [tex]$720^{\circ}$[/tex], and so on.
- This means to find the negative equivalent of [tex]$85^{\circ}$[/tex], we can subtract multiples of [tex]$360^{\circ}$[/tex] from [tex]$85^{\circ}$[/tex] until we get a negative angle.
2. Calculation:
- Subtract [tex]$360^{\circ}$[/tex] from [tex]$85^{\circ}$[/tex]:
[tex]\[ 85^{\circ} - 360^{\circ} = -275^{\circ} \][/tex]
- This tells us that [tex]$-275^{\circ}$[/tex] is one equivalent negative angle for [tex]$85^{\circ}$[/tex].
- We can also subtract another multiple of [tex]$360^{\circ}$[/tex]:
[tex]\[ 85^{\circ} - 720^{\circ} = -635^{\circ} \text{ (and so on)} \][/tex]
3. Verify Each Choice:
- Choice A: [tex]$-85^{\circ}$[/tex]
- Check [tex]\( -85^{\circ} = 85^{\circ} - 360^{\circ} \)[/tex]:
[tex]\[ 85^{\circ} - 360^{\circ} \neq -85^{\circ} \][/tex]
- [tex]\( -85^{\circ} \)[/tex] is not equivalent to [tex]\( 85^{\circ} \)[/tex].
- Choice B: [tex]$-265^{\circ}$[/tex]
- Check [tex]\( -265^{\circ} \)[/tex]:
[tex]\[ -265^{\circ} \text { as equivalent to another angle } = 85^{\circ} - 360^{\circ} * x \][/tex]
- Adding [tex]\( 360^{\circ} \)[/tex]:
[tex]\[ -265^{\circ} + 360^{\circ} = 95^{\circ} \][/tex]
- This is not equivalent to [tex]\( 85^{\circ} \)[/tex].
- Choice C: [tex]$-95^{\circ}$[/tex]
- Check [tex]\( -95^{\circ} \)[/tex]:
[tex]\[ before calculation, it seems irrelevant. - \[ 85^{\circ} - 360^{\circ} \neq -95^{\circ} \][/tex]
- Choice D: [tex]$-275^{\circ}$[/tex]
- Check [tex]\( -275^{\circ} \)[/tex]:
[tex]\[ 85^{\circ} - 360^{\circ} \][/tex]
- Correct, one full rotation leads to [tex]\( 85^{\circ} - 360^{\circ} = -275^{\circ} \)[/tex].
Therefore, among the given options, the negative angle equivalent to [tex]\(85^{\circ}\)[/tex] is:
D. An angle measuring [tex]$-275^{\circ}$[/tex]
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