What is the definition of the tangent function in relation to a right triangle?

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In the context of a **right triangle**, the **tangent function** (often abbreviated as "tan") relates one of the non-right angles to the lengths of two specific sides of the triangle. Here's a detailed definition:

### **Definition:**
For a given **acute angle** $$ \theta $$ in a **right triangle**, the tangent of $$ \theta $$ is defined as the ratio of the **length of the side opposite** to $$ \theta $$ to the **length of the side adjacent** to $$ \theta $$.

Mathematically, this is expressed as:

$$
\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}
$$

### **Components Explained:**

1. **Opposite Side:** This is the side that does **not** form the angle $$ \theta $$. It is directly across from $$ \theta $$.

2. **Adjacent Side:** This is the side that forms the angle $$ \theta $$ **alongside** the right angle (90°). It is next to $$ \theta $$ but is **not** the hypotenuse.

3. **Hypotenuse:** Although not directly involved in the definition of tangent, it's the longest side of a right triangle, opposite the right angle.

### **Visual Representation:**

Consider a right triangle labeled as follows:

```
      |
      |\
      | \
      |  \
      |   \
      |    \
      |     \
      |______\
```

- Let $$ \theta $$ be one of the non-right angles.
- The **opposite side** is the side directly across from $$ \theta $$.
- The **adjacent side** is the side next to $$ \theta $$ that forms the right angle.
- The **hypotenuse** is the side opposite the right angle.

### **Example:**

Suppose in a right triangle, for angle $$ \theta $$:
- The side opposite $$ \theta $$ is **3 units** long.
- The side adjacent to $$ \theta $$ is **4 units** long.

Then,

$$
\tan(\theta) = \frac{3}{4}
$$

### **Applications:**

- **Solving Triangles:** Knowing the tangent of an angle allows you to find unknown side lengths when two sides are known.
- **Trigonometric Identities:** Tangent is a fundamental function in various trigonometric identities and equations.
- **Real-World Problems:** Used in fields like engineering, physics, architecture, and navigation to resolve components of forces, angles of elevation/depression, and more.

### **Important Notes:**

- The tangent function is **undefined** when the adjacent side is **0**, which occurs at angles of 90° and 270° in the unit circle context.
- Tangent can also be extended beyond right triangles using the unit circle and more advanced trigonometric definitions, but within the scope of right triangles, it's always the ratio of the opposite side to the adjacent side.

Understanding the tangent function in right triangles is fundamental to solving many geometric and real-world problems involving angles and distances.
In a right triangle, the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

What is the definition of the tangent function in relation to a right triangle?

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