To determine the coordinates of the terminal point determined by [tex]\( t = \frac{10 \pi}{3} \)[/tex], we need to consider the angle's position within the unit circle. Here's the step-by-step process:
1. Normalize the Angle:
Since [tex]\( t = \frac{10 \pi}{3} \)[/tex] is greater than [tex]\( 2 \pi \)[/tex], we must reduce it to be within the range of [tex]\( 0 \)[/tex] to [tex]\( 2 \pi \)[/tex]. To do this, we find the equivalent angle by subtracting multiples of [tex]\( 2 \pi \)[/tex].
[tex]\[
\frac{10 \pi}{3} \mod 2\pi = \frac{10 \pi}{3} - 2\pi \times \left\lfloor \frac{10 \pi / 3}{2 \pi} \right\rfloor
\][/tex]
Simplifying this, we get:
[tex]\[
\frac{10\pi}{3} \mod 2\pi = \frac{10\pi}{3} - 2\pi \times 1 = \frac{10\pi}{3} - \frac{6\pi}{3} = \frac{4\pi}{3}
\][/tex]
2. Calculate the Coordinates:
Now, we need to find the coordinates corresponding to [tex]\( \frac{4\pi}{3} \)[/tex].
The angle [tex]\( \frac{4\pi}{3} \)[/tex] is in the third quadrant, where both sine and cosine are negative. The unit circle coordinates for [tex]\( \frac{4\pi}{3} \)[/tex] are given by [tex]\( (\cos \frac{4\pi}{3}, \sin \frac{4\pi}{3}) \)[/tex].
[tex]\[
\cos \frac{4\pi}{3} = -\frac{1}{2}
\][/tex]
[tex]\[
\sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}
\][/tex]
3. Identify the Correct Choice:
Comparing these coordinates to the given choices:
A. [tex]\(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex]
B. [tex]\(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)[/tex]
C. [tex]\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)[/tex]
D. [tex]\(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex]
We see that the coordinates [tex]\( (\cos \frac{4\pi}{3}, \sin \frac{4\pi}{3}) = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \)[/tex] match choice C.
Hence, the coordinates of the terminal point determined by [tex]\( t = \frac{10 \pi}{3} \)[/tex] are:
[tex]\[
\boxed{\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)}
\][/tex]
