To find the coordinates of the terminal point determined by [tex]\( t = \frac{10\pi}{3} \)[/tex], we need to follow these steps:
1. Identify the coterminal angle:
Since the given angle [tex]\( t \)[/tex] is more than [tex]\( 2\pi \)[/tex], we need to find a coterminal angle within the interval [tex]\([0, 2\pi)\)[/tex]. This can be done by subtracting [tex]\( 2\pi \)[/tex] from [tex]\( \frac{10\pi}{3} \)[/tex]:
[tex]\[
\frac{10\pi}{3} - 2\pi = \frac{10\pi}{3} - \frac{6\pi}{3} = \frac{4\pi}{3}
\][/tex]
The coterminal angle is [tex]\( \frac{4\pi}{3} \)[/tex].
2. Determine the quadrant:
The angle [tex]\( \frac{4\pi}{3} \)[/tex] is greater than [tex]\( \pi \)[/tex] but less than [tex]\( 3\pi/2 \)[/tex]. This places it in the third quadrant.
3. Calculate the reference angle:
The reference angle in the third quadrant is found by subtracting [tex]\( \pi \)[/tex] from [tex]\( \frac{4\pi}{3} \)[/tex]:
[tex]\[
\frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3}
\][/tex]
So, the reference angle is [tex]\( \frac{\pi}{3} \)[/tex].
4. Find the coordinates using the unit circle:
The coordinates corresponding to the reference angle [tex]\( \frac{\pi}{3} \)[/tex] are [tex]\( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \)[/tex].
However, because we are in the third quadrant, both the x and y coordinates need to be negative:
[tex]\[
\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)
\][/tex]
Therefore, the coordinates of the terminal point [tex]\( t = \frac{10\pi}{3} \)[/tex] are:
A. [tex]\( \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \)[/tex]
So, the correct answer is [tex]\( \boxed{\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)} \)[/tex].
