Question 1170877: Consider the following.
t=-2pi/3
(a) Find the reference number t for the value of t.
t =
(b) Find the terminal point determined by t.
(x, y) =
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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)
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