Draw the angle in standard position.  State the quadrant in which the angle lies.  Work the exercise without converting to degrees.

negative StartFraction 5 pi Over 3 EndFraction−5π3

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Part 1

Choose the correct graph below.

A.

B.

C.

D.

Part 2

The angle is in quadrant 

I.

IV.

II.

III.

I.

Let's break this down step by step to determine the graph and the quadrant for the angle \( -\frac{5\pi}{3} \).

### Step 1: Determine the angle's direction
The angle is negative, which means it is measured clockwise starting from the positive x-axis. To understand where this angle lies, we need to find a positive coterminal angle.

### Step 2: Find the positive coterminal angle
A coterminal angle is found by adding \( 2\pi \) (which is a full circle) to the given angle. In this case, we'll add \( 2\pi \) to \( -\frac{5\pi}{3} \):

\[
- \frac{5\pi}{3} + 2\pi = -\frac{5\pi}{3} + \frac{6\pi}{3} = \frac{\pi}{3}
\]

So, the positive coterminal angle is \( \frac{\pi}{3} \).

### Step 3: Identify the quadrant
Since \( \frac{\pi}{3} \) is a positive angle less than \( \frac{\pi}{2} \), it lies in **Quadrant I**.

However, the original angle \( -\frac{5\pi}{3} \) is a negative rotation. We can conclude that it wraps clockwise and lands in **Quadrant IV**, because adding the full circle leads us to a coterminal angle in Quadrant I.

### Part 2: The angle is in **Quadrant IV**.

Would you like to see a visual representation or further clarifications?

### Five Related Questions:
1. What is a coterminal angle, and how is it calculated?
2. How do you convert between radians and degrees?
3. What is the significance of the reference angle in quadrant determination?
4. How do you determine the quadrant of a negative angle?
5. What is the relationship between positive and negative angles?

**Tip:** Always check for coterminal angles to simplify the quadrant determination process!

Learn how to draw the angle -5π/3 in standard position and determine its quadrant without converting to degrees. Understand the concept of coterminal angles and their role in quadrant determination. Suitable for high school students studying trigonometry and angle measurement.

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