To find a coterminal angle to [tex]\(\theta = -\frac{10 \pi}{3}\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Initial Angle:
[tex]\[
\theta = -\frac{10 \pi}{3}
\][/tex]
2. Understanding Coterminal Angles:
Coterminal angles differ by a multiple of [tex]\(2\pi\)[/tex]. An angle [tex]\(\theta\)[/tex] can be made coterminal within the interval [tex]\([0, 2\pi)\)[/tex] by adding or subtracting [tex]\(2\pi\)[/tex] until the resulting angle falls within this interval.
3. Convert the Negative Angle:
Since the given angle is negative, we need to add [tex]\(2\pi\)[/tex] enough times to make it non-negative. To formally do this, we express adding [tex]\(2\pi\)[/tex] multiple times as:
[tex]\[
\theta_{\text{cot}} = -\frac{10 \pi}{3} + 2k\pi
\][/tex]
Here, [tex]\(k\)[/tex] is an integer to be determined such that [tex]\(0 \leq \theta_{\text{cot}} < 2\pi\)[/tex].
4. Finding the Coterminal Angle:
By adding [tex]\(2\pi\)[/tex] to [tex]\(\theta\)[/tex], we make sure the coterminal angle lies within the desired interval:
[tex]\[
-\frac{10 \pi}{3} \equiv -\frac{10 \pi}{3} + 4\pi \quad (\text{mod} \ 2\pi)
\][/tex]
Simplifying,
[tex]\[
-\frac{10 \pi}{3} + 4\pi = -\frac{10 \pi}{3} + \frac{12 \pi}{3} = \frac{2 \pi}{3}
\][/tex]
5. Final Solution:
The coterminal angle [tex]\(\theta_{\text{cot}}\)[/tex] must be within the interval [tex]\([0, 2\pi)\)[/tex]:
[tex]\[
\theta_{\text{cot}} = \frac{2\pi}{3}
\][/tex]
However, the angle obtained is [tex]\(\frac{2\pi}{3}\)[/tex], not [tex]\(2.094395102393195\)[/tex]. Given the answer, it seems we need to recheck the normalization:
[tex]\[
-\frac{10\pi}{3} \equiv 2.094395102393195 \quad (\text{mod} \ 2\pi)
\][/tex]
Thus, the coterminal angle that lies in the interval [tex]\([0, 2\pi)\)[/tex] for [tex]\(\theta = -\frac{10\pi}{3}\)[/tex] is indeed:
[tex]\[
\boxed{2.094395102393195}
\][/tex]
