Question

Explain trigonometry to me
8th grade level
Formulas and examples
Easy

Answer 1

Sure! Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, especially right triangles (triangles with one angle measuring 90 degrees). Let’s break it down in a simple way.

### Basic Vocabulary:

1. **Triangle**: A shape with three sides and three angles.
2. **Right Triangle**: A triangle with one angle that is exactly 90 degrees.
3. **Hypotenuse**: The longest side of a right triangle, opposite the right angle.
4. **Adjacent Side**: The side that is next to (or "adjacent" to) a given angle.
5. **Opposite Side**: The side that is opposite to a given angle.

### Key Trigonometric Ratios:

There are three main ratios in trigonometry that help us understand the relationships between the angles and sides of a right triangle:

1. **Sine (sin)**:
- Formula: \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
- Example: In a right triangle where the angle θ is 30 degrees, if the hypotenuse is 10 units long, the side opposite to θ might be 5 units. Then, \( \sin(30°) = \frac{5}{10} = 0.5 \).

2. **Cosine (cos)**:
- Formula: \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
- Example: Using the same triangle, if the adjacent side (next to θ) is \( \sqrt{75} \) units, then \( \cos(30°) = \frac{\sqrt{75}}{10} \).

3. **Tangent (tan)**:
- Formula: \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)
- Example: In our triangle, if the opposite side is still 5 units and the adjacent side is approximately 8.66 units, then \( \tan(30°) = \frac{5}{8.66} \approx 0.577 \).

### Remembering the Ratios:

You can use the acronym **SOH-CAH-TOA** to remember the definitions:

- **SOH**: Sine = Opposite / Hypotenuse
- **CAH**: Cosine = Adjacent / Hypotenuse
- **TOA**: Tangent = Opposite / Adjacent

### Example Problem:

Imagine you have a right triangle where one angle is 45 degrees, the hypotenuse is 10 units, and you need to find the lengths of the opposite and adjacent sides.

1. **Using Sine**:
- \( \sin(45°) = \frac{\text{Opposite}}{10} \)
- Since \( \sin(45°) \) is approximately 0.707,
- \( 0.707 = \frac{\text{Opposite}}{10} \)
- Thus, \( \text{Opposite} ≈ 0.707 \times 10 ≈ 7.07 \) units.

2. **Using Cosine**:
- \( \cos(45°) = \frac{\text{Adjacent}}{10} \)
- This will give the same calculation as for the opposite side because of the 45-degree angle in a right triangle.
- So \( \text{Adjacent} ≈ 7.07 \) units too.

### Conclusion:

Trigonometry helps us calculate unknown sides or angles in a triangle using these ratios. Understanding these basic concepts allows you to solve many practical problems related to angles and distances. With practice, you'll become more comfortable using these formulas!