TR-38: Angle Sum and Difference Identities (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial provides a comprehensive guide on Angle Sum and Difference Identities in trigonometry, based on the animated proofs and explanations presented in Dennis F. Davis's video. Understanding these identities is essential for solving various trigonometric problems, particularly in advanced mathematics courses and exams like the International A Level.

Step 1: Understanding Angle Sum Identities

Angle Sum Identities help you find the sine, cosine, and tangent of the sum of two angles. Here are the key identities to remember:

• Sine Identity

  • sin(A + B) = sin(A) * cos(B) + cos(A) * sin(B)

• Cosine Identity

  • cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)

• Tangent Identity

  • tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) * tan(B))

Practical Advice

• Familiarize yourself with these identities, as they form the foundation for solving trigonometric equations.
• Practice using these identities with different angle values to build confidence.

Step 2: Understanding Angle Difference Identities

Similar to the sum identities, the Angle Difference Identities help calculate the sine, cosine, and tangent of the difference between two angles:

• Sine Identity

  • sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B)

• Cosine Identity

  • cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)

• Tangent Identity

  • tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A) * tan(B))

Practical Advice

• Memorizing these identities alongside the sum identities will help you solve problems that involve both addition and subtraction of angles.
• Use diagrams to visualize how these identities work, enhancing your understanding.

Step 3: Application of Identities in Problems

To effectively use Angle Sum and Difference Identities, let's go through a few examples:

1. Example Problem for Sine Identity

   • Find sin(75°) using sin(45° + 30°):
     • sin(75°) = sin(45°) * cos(30°) + cos(45°) * sin(30°)
     • Substitute known values:
       • sin(45°) = √2/2
       • cos(30°) = √3/2
       • cos(45°) = √2/2
       • sin(30°) = 1/2
     • Calculate:
       • sin(75°) = (√2/2 * √3/2) + (√2/2 * 1/2) = (√6/4) + (√2/4) = (√6 + √2)/4

2. Example Problem for Cosine Identity

   • Find cos(105°) using cos(60° + 45°):
     • cos(105°) = cos(60°) * cos(45°) - sin(60°) * sin(45°)
     • Substitute known values:
       • cos(60°) = 1/2
       • cos(45°) = √2/2
       • sin(60°) = √3/2
       • sin(45°) = √2/2
     • Calculate:
       • cos(105°) = (1/2 * √2/2) - (√3/2 * √2/2) = (√2/4) - (√6/4) = (√2 - √6)/4

Common Pitfalls

• Be careful with signs in the identities when using angles in different quadrants.
• Always check if angles can be simplified before applying the identities.

Conclusion

In this tutorial, you learned about Angle Sum and Difference Identities, their definitions, applications, and example problems. Mastering these identities is crucial for tackling more complex trigonometric challenges. For further practice, try using these identities on various angle combinations or explore their applications in real-world scenarios such as physics and engineering.

Next steps include practicing problems regularly and exploring additional trigonometric concepts to deepen your understanding.