🌿🌿👉👉QUADRANT RULES👈👈💚💚

 The **Quadrant Rule** in math refers to the system used to divide the **coordinate plane** into four sections, called **quadrants**, based on the signs of the **x** and **y** coordinates. This is essential in **graphing points, analyzing functions, and trigonometry**.

---

### **The Four Quadrants**

The **xy-plane** is divided by the **x-axis** (horizontal) and **y-axis** (vertical) into four quadrants, labeled **I, II, III, and IV** in **counterclockwise** order:

| Quadrant | \(x\)-coordinate | \(y\)-coordinate | Example Point |

|----------|------------------|------------------|---------------|

| **I**    | Positive (+)     | Positive (+)     | \((3, 4)\)     |

| **II**   | Negative (−)     | Positive (+)     | \((-2, 5)\)    |

| **III**  | Negative (−)     | Negative (−)     | \((-1, -3)\)   |

| **IV**   | Positive (+)     | Negative (−)     | \((6, -2)\)    |

- **Axes:** Points on the x-axis (\(y=0\)) or y-axis (\(x=0\)) do not belong to any quadrant.

---

### **Key Rules & Applications**

1. **Signs of Coordinates**  

   - **Quadrant I:** \((+, +)\)  

   - **Quadrant II:** \((-, +)\)  

   - **Quadrant III:** \((-, -)\)  

   - **Quadrant IV:** \((+, -)\)

2. **Trigonometric Functions (Unit Circle)**  

   - Angles are measured from the positive x-axis.  

   - The signs of **sine, cosine, and tangent** depend on the quadrant:

     - **Q1:** All \((sin, cos, tan)\) are positive.  

     - **Q2:** Only \(sin\) is positive.  

     - **Q3:** Only \(tan\) is positive.  

     - **Q4:** Only \(cos\) is positive.  

   *(Mnemonic: **"All Students Take Calculus"** for A, S, T, C.)*

3. **Graphing Equations**  

   - The quadrant affects the behavior of functions:  

     - **Linear equations:** Slope determines direction.  

     - **Parabolas:** Vertex position changes based on coefficients.  

4. **Distance & Midpoint Formulas**  

   - Calculations remain the same, but signs matter for quadrant placement.  

---

### **Example Problems**

1. **Identifying Quadrants:**  

   - \((−5, 1)\) → **Quadrant II** (since \(x < 0, y > 0\)).  

   - \((4, −7)\) → **Quadrant IV** (since \(x > 0, y < 0\)).

2. **Trigonometry Example:**  

   - If an angle \(θ\) is in **Quadrant III**, and \(\sin θ = -\frac{3}{5}\), then \(\cos θ\) is also negative (since only \(tan\) is positive in Q3).

---

### **Special Cases**

- **Origin:** \((0, 0)\) is where the axes meet (no quadrant).  

- **Boundaries:** Points like \((3, 0)\) lie on the x-axis, not in a quadrant.

---

### **Visualization**

```

          y

          |

   II     |     I

          |

------------------> x

          |

  III     |     IV

          |

```

Let me know if you'd like deeper explanations on quadrants in graphing, trigonometry, or other applications! 😊

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  🌿🌿👉👉👉THANK YOU👈👈🌿🌿♥️♥️