## Trigonometry: A Comprehensive Guide for GCSE Mathematics

### Meta Description
Master trigonometry for GCSE Maths with this complete guide. Learn key concepts, practice problems, and exam tips to excel in your exams.

### Introduction
**What is Trigonometry?**
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is essential in GCSE Mathematics, used in various applications such as navigation, architecture, and engineering.

**Why is it important in GCSE Mathematics?**
Trigonometry enables students to:
- Calculate missing angles and sides in triangles
- Solve real-world problems involving angles and distances
- Understand trigonometric functions and their applications

**Real-World Applications**
Trigonometry is used in a wide range of fields, including:
- Surveying: Calculating distances between objects
- Navigation: Determining the direction and distance of travel
- Construction: Designing structures and calculating angles

### Main Content
**Key Concepts and Definitions**
- **Sine (sin):** The ratio of the opposite side to the hypotenuse of a right-angled triangle.
- **Cosine (cos):** The ratio of the adjacent side to the hypotenuse of a right-angled triangle.
- **Tangent (tan):** The ratio of the opposite side to the adjacent side of a right-angled triangle.
- **Pythagoras' Theorem:** a² + b² = c², where a and b are the lengths of the shorter sides and c is the length of the hypotenuse of a right-angled triangle.

**Step-by-Step Explanations**
- **Finding missing angles:** Use the sine, cosine, or tangent ratios to calculate the unknown angle.
- **Calculating missing sides:** Use Pythagoras' Theorem to find the missing side length.
- **Using trigonometric functions:** Understand how to use the sin, cos, and tan functions to solve problems.

**Common Mistakes to Avoid**
- Mixing up sin, cos, and tan
- Using the wrong side lengths in calculations
- Forgetting to convert angles to radians (when necessary)

### Practice Problems
**Example 1:**
Find the length of the missing side of a right-angled triangle with an adjacent side of 5 cm and an opposite side of 12 cm.
**Solution:**
Use Pythagoras' Theorem: 5² + 12² = c². Solve for c: c = 13 cm.

**FAQ**
**Q1: How do I remember the trigonometric ratios?**
* **A:** Use the acronym SOH CAH TOA: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), Tangent (Opposite/Adjacent).
**Q2: When do I need to use radians instead of degrees?**
* **A:** Radians are used in many trigonometry equations, such as when using the sine, cosine, and tangent functions to calculate exact values.

### Conclusion
**Summary of Key Points**
- Trigonometry involves the relationships between sides and angles in triangles.
- Sine, cosine, and tangent are trigonometric ratios used to calculate missing angles and sides.
- Pythagoras' Theorem is used to find the missing side length of a right-angled triangle.

**Tips for Exam Success**
- Practice regularly with different types of problems.
- Understand the concepts behind trigonometric ratios and Pythagoras' Theorem.
- Convert angles to radians when necessary for accurate calculations.

**Links to Practice Resources**
- Khan Academy Trigonometry: https://www.khanacademy.org/math/trigonometry
- GCSE Maths Tutor Trigonometry: https://www.gcsemathstutor.com/Trigonometry.html