Find the terminal point
P(x, y)
on the unit circle determined by the given value of t.
t = −
4𝜋
3

To find the terminal point \( P(x, y) \) on the unit circle for \( t = -\frac{4\pi}{3} \), we follow these steps:

1. **Identify the reference angle**:
   \[
   -\frac{4\pi}{3} \quad \text{corresponds to an angle of} \quad \frac{4\pi}{3} \quad \text{in the positive direction}
   \]
   since rotating by \( -\frac{4\pi}{3} \) is equivalent to rotating in the opposite (clockwise) direction.

2. **Locate the terminal angle**:
   The angle \( \frac{4\pi}{3} \) in standard position is in the third quadrant because it is greater than \( \pi \) but less than \( \frac{3\pi}{2} \).

3. **Calculate coordinates on the unit circle**:
   - For the unit circle, the coordinates \( (x, y) \) for an angle of \( \theta = \frac{4\pi}{3} \) are given by:
     \[
     x = \cos\left(\frac{4\pi}{3}\right) \quad \text{and} \quad y = \sin\left(\frac{4\pi}{3}\right)
     \]
   - Since \( \frac{4\pi}{3} \) is in the third quadrant, both \( \cos\left(\frac{4\pi}{3}\right) \) and \( \sin\left(\frac{4\pi}{3}\right) \) are negative.

4. **Compute the exact values**:
   \[
   \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \quad \text{and} \quad \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}
   \]

Thus, the terminal point \( P(x, y) \) is:
\[
P\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)
\]

Would you like further details on any part of this solution, or do you have any questions?

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Here are five related questions:
1. How would you find the terminal point for \( t = \frac{5\pi}{3} \)?
2. What is the terminal point for \( t = \pi \) on the unit circle?
3. How does one determine which quadrant a terminal point is in?
4. What are the coordinates of the terminal point for \( t = -\frac{\pi}{6} \)?
5. How do reference angles help in finding terminal points?

**Tip:** The coordinates \( (\cos(t), \sin(t)) \) always give the terminal point on the unit circle for any angle \( t \).

Find the terminal point P(x, y) on the unit circle determined by the given value of t. t = -4π/3

Explore how to find the terminal point P(x, y) on the unit circle for t = -4π/3. This trigonometry problem covers key concepts such as reference angles and coordinates on the unit circle, providing a step-by-step solution suitable for Grades 9-11.

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