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What rotation about the origin is equivalent to [tex]$R_{-500^{\circ}}$[/tex]?

A. [tex][tex]$R_{280^{\circ}}$[/tex][/tex]
B. [tex]$R_{220^{\circ}}$[/tex]
C. [tex]$R_{-220^{\circ}}$[/tex]
D. [tex][tex]$R_{-280^{\circ}}$[/tex][/tex]


Sagot :

To find an equivalent rotation angle when given a negative rotation of [tex]\(-500^{\circ}\)[/tex], we first need to normalize this angle within the range [tex]\(0^{\circ}\)[/tex] to [tex]\(360^{\circ}\)[/tex].

Here's how we can do this step-by-step:

1. Understanding the Problem:
We need an equivalent angle to [tex]\(-500^{\circ}\)[/tex] that lies between [tex]\(0^{\circ}\)[/tex] and [tex]\(360^{\circ}\)[/tex].

2. Normalization Step:
Normalize the angle by using the modulo operation with [tex]\(360^{\circ}\)[/tex].

[tex]\[ \text{normalized\_angle} = -500 \mod 360 \][/tex]

3. Calculating the Normalized Angle:
The modulo operation essentially tells us the remainder when [tex]\(-500\)[/tex] is divided by [tex]\(360\)[/tex]. But given it's negative, it's useful to visualize the operation:

- Start from [tex]\(-500\)[/tex]
- Add [tex]\(360\)[/tex] repeatedly until the result is between [tex]\(0^{\circ}\)[/tex] and [tex]\(360^{\circ}\)[/tex].

[tex]\[ -500 + 360 = -140 \][/tex]
[tex]\[ -140 + 360 = 220 \][/tex]

So, the equivalent positive angle is [tex]\(220^{\circ}\)[/tex].

4. Final Equivalent Angle:
The angle [tex]\(220^{\circ}\)[/tex] is within the range [tex]\(0^{\circ}\)[/tex] to [tex]\(360^{\circ}\)[/tex], and it represents the same rotation about the origin as [tex]\(-500^{\circ}\)[/tex] but in the positive direction.

Therefore, the equivalent rotation about the origin for [tex]\(-500^{\circ}\)[/tex] is:

[tex]\[ \boxed{220^{\circ}} \][/tex]

Conclusion:

[tex]\[ \text{Option B. } R_{220} \][/tex]