To find the terminal point on the unit circle determined by an angle of [tex]\(\frac{4\pi}{3}\)[/tex] radians, we need to determine the coordinates [tex]\((x, y)\)[/tex] at this angle. The unit circle has a radius of 1, which implies that any point on the circle will satisfy the equation [tex]\(x^2 + y^2 = 1\)[/tex].
### Step-by-Step Solution:
1. Determine the angle on the unit circle:
The angle [tex]\(\frac{4\pi}{3}\)[/tex] radians is more than [tex]\(\pi\)[/tex] (180 degrees) but less than [tex]\(2\pi\)[/tex] (360 degrees). Since [tex]\(\frac{4\pi}{3} = \pi + \frac{\pi}{3}\)[/tex], we can see that this angle is [tex]\(\pi/3\)[/tex] radians past [tex]\(\pi\)[/tex].
2. Locate the angle in the correct quadrant:
- The angle [tex]\(\pi\)[/tex] radians is 180 degrees, which is the negative [tex]\(x\)[/tex]-axis.
- Adding [tex]\(\pi/3\)[/tex] to [tex]\(\pi\)[/tex] places the terminal point in the third quadrant.
3. Reference angle:
The reference angle for [tex]\(\frac{4\pi}{3}\)[/tex] is [tex]\(\pi/3\)[/tex].
4. Cosine and Sine values for the reference angle:
- [tex]\(\cos(\pi/3) = \frac{1}{2}\)[/tex]
- [tex]\(\sin(\pi/3) = \frac{\sqrt{3}}{2}\)[/tex]
5. Apply signs based on the quadrant:
In the third quadrant:
- The cosine value is negative.
- The sine value is also negative.
Thus:
- [tex]\(\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}\)[/tex]
- [tex]\(\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)[/tex]
6. Determine the coordinates:
Based on the values from above, the coordinates [tex]\((x, y)\)[/tex] are:
- [tex]\(x = \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}\)[/tex]
- [tex]\(y = \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)[/tex]
Therefore, the terminal point on the unit circle determined by the angle [tex]\(\frac{4\pi}{3}\)[/tex] radians is [tex]\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)[/tex].
