Question 1170877
You've asked this question before. Here's the solution again:

**Given:**

* t = -2π/3

**(a) Find the reference number t' for the value of t.**

The reference number t' is the shortest distance along the unit circle from the terminal point determined by t to the x-axis.

1. **Determine the Quadrant:**
   * -2π/3 is in the third quadrant.
   * To see this, note that -π is -3π/3, and -π/2 is -1.5π/3. -2π/3 lies between these values.

2. **Calculate the Reference Angle:**
   * Since t is in the third quadrant, the reference angle is calculated as t' = |t - (-π)|.
   * t' = |-2π/3 - (-π)| = |-2π/3 + π| = |-2π/3 + 3π/3| = |π/3| = π/3

Therefore, the reference number t' is π/3.

**(b) Find the terminal point (x, y) determined by t.**

1. **Use the Reference Angle:**
   * The reference angle is π/3.
   * The coordinates for π/3 on the unit circle are (1/2, √3/2).

2. **Adjust for the Quadrant:**
   * Since t = -2π/3 is in the third quadrant, both x and y coordinates are negative.
   * Therefore, the terminal point is (-1/2, -√3/2).

**Answers:**

(a) t' = π/3
(b) (x, y) = (-1/2, -√3/2)