TR-13Z: How the Co- Trig Functions got their Names

Introduction

This tutorial explains how the co-trigonometric functions—cosine, cotangent, and cosecant—acquired their 'co-' prefix. Understanding the origins of these terms can enhance your grasp of trigonometry, particularly in contexts like A-level mathematics or various exam boards such as Edexcel and Cambridge.

Step 1: Understanding Trigonometric Functions

Before delving into the co-functions, it's essential to familiarize yourself with the basic trigonometric functions:

• Sine (sin): Opposite side over hypotenuse.
• Cosine (cos): Adjacent side over hypotenuse.
• Tangent (tan): Opposite side over adjacent side.

Practical Advice

• Remember the relationships between these functions as they form the basis for understanding their co-functions.

Step 2: Introduction to Co-Functions

Co-functions are pairs of trigonometric functions that relate to each other in a specific way. The co-functions are:

• Cosine (cos) and Sine (sin)
• Cotangent (cot) and Tangent (tan)
• Cosecant (csc) and Secant (sec)

Explanation of Co-Functions

The term 'co-' indicates a complementary relationship. For example:

• cos(θ) = sin(90° - θ)
• cot(θ) = tan(90° - θ)
• csc(θ) = sec(90° - θ)

Practical Advice

• Use the complementary angles to derive values of co-functions easily.

Step 3: The Origin of the 'Co-' Prefix

The prefix 'co-' originates from the concept of complementary angles in trigonometry. Each co-function can be defined in relation to its primary function using 90 degrees (or π/2 radians).

Key Points

• The cosine function represents the sine of the complementary angle.
• The cotangent function represents the tangent of the complementary angle.
• The cosecant function represents the secant of the complementary angle.

Step 4: Application of Co-Functions

Understanding co-functions is crucial in solving trigonometric equations and in various applications, such as:

• Simplifying expressions in calculus.
• Solving triangles in geometry.
• Analyzing periodic functions in physics.

Practical Tips

• When solving problems, always consider the complementary relationships to simplify calculations.
• Familiarize yourself with the unit circle, which helps visualize these relationships.

Conclusion

The co-trigonometric functions offer a valuable perspective in trigonometry by illustrating the relationships between angles. By understanding how cosine, cotangent, and cosecant derive their names and their functions, you can enhance your mathematical skills and problem-solving abilities. Consider practicing problems that involve these relationships to solidify your understanding and prepare for exams.