TR-15Z: Proof of the Common Sine and Cosine Values

Introduction

This tutorial aims to provide a comprehensive proof of the sine and cosine values for common angles, specifically 30°, 45°, and 60°. Understanding these values is essential for higher-level mathematics, especially in trigonometry, and is relevant for various exams, including A level and International A level certifications.

Step 1: Understanding the Unit Circle

The unit circle is fundamental in deriving sine and cosine values.

• The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.
• The coordinates of any point on the circle can be expressed as (cos(θ), sin(θ)), where θ is the angle measured from the positive x-axis.
• Familiarize yourself with the placement of common angles on the unit circle:
  • 30° (π/6 radians)
  • 45° (π/4 radians)
  • 60° (π/3 radians)

Step 2: Deriving Sine and Cosine for 30° and 60°

To find the sine and cosine values for 30° and 60°, you can use the properties of special triangles.

• 30° Triangle (30-60-90 Triangle)

  • The sides are in the ratio 1:√3:2.
  • For 30°:
    • sin(30°) = opposite/hypotenuse = 1/2
    • cos(30°) = adjacent/hypotenuse = √3/2

• 60° Triangle

  • The same triangle can be used:
  • For 60°:
    • sin(60°) = opposite/hypotenuse = √3/2
    • cos(60°) = adjacent/hypotenuse = 1/2

Step 3: Deriving Sine and Cosine for 45°

The sine and cosine values for 45° can be derived from a 45-45-90 triangle.

• 45° Triangle
  • The sides are equal, so if each leg is 1, the hypotenuse is √2.
  • For 45°:
    • sin(45°) = opposite/hypotenuse = 1/√2 = √2/2
    • cos(45°) = adjacent/hypotenuse = 1/√2 = √2/2

Step 4: Summary of Common Sine and Cosine Values

Here’s a quick reference for the sine and cosine values of the common angles:

• sin(30°) = 1/2
• cos(30°) = √3/2
• sin(45°) = √2/2
• cos(45°) = √2/2
• sin(60°) = √3/2
• cos(60°) = 1/2

Conclusion

Understanding and deriving sine and cosine values for common angles is crucial for solving trigonometric problems. Use the unit circle and special triangles as tools to memorize these values. For further practice, try solving trigonometric equations or applying these values to real-world scenarios, such as physics problems involving angles. Consider exploring additional resources or videos to deepen your understanding.