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]
