To find the coordinates of the terminal point for the angle [tex]\( t = \frac{10 \pi}{3} \)[/tex], we can proceed with a step-by-step approach:
1. Normalize the Angle:
The angle given is [tex]\( \frac{10 \pi}{3} \)[/tex], which is greater than [tex]\( 2\pi \)[/tex]. To determine the equivalent angle within one full circle (between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex]), we need to reduce this angle by subtracting multiples of [tex]\(2\pi\)[/tex].
[tex]\[
\frac{10\pi}{3} \mod 2\pi = \frac{10\pi}{3} - 2\pi \left\lfloor \frac{\frac{10\pi}{3}}{2\pi} \right\rfloor
\][/tex]
First, we calculate the integer part of [tex]\( \frac{\frac{10\pi}{3}}{2\pi} \)[/tex]:
[tex]\[
\frac{10\pi}{3} \div 2\pi = \frac{10\pi}{3} \cdot \frac{1}{2\pi} = \frac{10}{6} = \frac{5}{3} \approx 1.6667
\][/tex]
So, the integer part is [tex]\(1\)[/tex]. Now we subtract this from the original angle:
[tex]\[
\frac{10\pi}{3} - 2\pi \times 1 = \frac{10\pi}{3} - \frac{6\pi}{3} = \frac{4\pi}{3}
\][/tex]
2. Calculate Coordinates:
The reduced angle is [tex]\( \frac{4\pi}{3} \)[/tex], and now, we need to find the [tex]\((x, y)\)[/tex] coordinates on the unit circle for this angle. The unit circle coordinates for any angle [tex]\( t \)[/tex] are given by:
[tex]\[
x = \cos(t)
\][/tex]
[tex]\[
y = \sin(t)
\][/tex]
For [tex]\( t = \frac{4\pi}{3} \)[/tex]:
[tex]\[
\cos(\frac{4\pi}{3}) \approx -0.5
\][/tex]
[tex]\[
\sin(\frac{4\pi}{3}) \approx -\frac{\sqrt{3}}{2}
\][/tex]
Therefore, the coordinates of the terminal point for [tex]\( t = \frac{10 \pi}{3} \)[/tex] are:
[tex]\[
\left(-0.5, -\frac{\sqrt{3}}{2}\right)
\][/tex]
3. Conclusion:
The coordinates of the terminal point are [tex]\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)[/tex], which corresponds to option
[tex]\[
\boxed{\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)}
\][/tex]
