To find the reduced representation in radians of the expression [tex]\(\operatorname{rr}\left(\frac{250 \pi}{3}\right)\)[/tex], we need to determine the equivalent angle within the standard interval [tex]\( [0, 2\pi) \)[/tex].
Given the value:
[tex]\[ \frac{250\pi}{3} \][/tex]
1. Understand the Concept: The reduced representation essentially asks us to convert this angle into an equivalent angle between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex]. This means we need to find the remainder when [tex]\(\frac{250\pi}{3}\)[/tex] is divided by [tex]\(2\pi\)[/tex].
2. Calculate the Remainder:
- We need to determine [tex]\(\frac{250\pi}{3} \mod 2\pi\)[/tex].
3. Simplify the Expression:
- Since [tex]\(2\pi\)[/tex] is the period of a full circle in radians, we subtract or add multiples of [tex]\(2\pi\)[/tex] until we get an angle within the desired range.
Instead of performing this calculation manually, let's refer to the mathematically derived result:
[tex]\[ \frac{250\pi}{3} \mod 2\pi = 4.188790204786393 \][/tex]
So, the reduced representation in radians of [tex]\(\frac{250 \pi}{3}\)[/tex] is:
[tex]\[ \operatorname{rr}\left(\frac{250 \pi}{3}\right) = 4.188790204786393 \][/tex]
This value is within the interval [tex]\( [0, 2\pi) \)[/tex]. Thus, the angle's reduced representation, or its equivalent angle in the standard interval, is approximately [tex]\( 4.189 \)[/tex] radians when rounded to three decimal places.
