TR-40: Double Angle Identities (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial focuses on the Double Angle Identities in trigonometry, specifically for sine and cosine functions. Understanding these identities is essential for solving various trigonometric problems and is relevant for students preparing for exams like the International A Level and Edexcel. By the end of this guide, you will be able to prove and apply these identities effectively.

Step 1: Understanding Double Angle Identities

Double Angle Identities express trigonometric functions of double angles in terms of single angles. The key identities to memorize are:

• For sine:

  • sin(2θ) = 2sin(θ)cos(θ)

• For cosine:

  • cos(2θ) = cos²(θ) - sin²(θ)
  • Alternatively, this can also be expressed as:
    • cos(2θ) = 2cos²(θ) - 1
    • cos(2θ) = 1 - 2sin²(θ)

Practical Tip

To remember these identities, note that the sine identity involves a multiplication of sine and cosine, while the cosine identity can be expressed in three different forms, giving you flexibility based on what you need in your calculations.

Step 2: Proving the Double Angle Identities

Proving the identities can help solidify your understanding. Here’s how to prove each identity:

Proving sin(2θ)

1. Start with the left side: sin(2θ).
2. Use the angle addition formula:
   • sin(2θ) = sin(θ + θ)
3. Apply the formula:
   • sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ)
4. Combine like terms:
   • sin(2θ) = 2sin(θ)cos(θ)

Proving cos(2θ)

1. Start with cos(2θ).

2. Use the angle addition formula:

   • cos(2θ) = cos(θ + θ)

3. Apply the formula:

   • cos(θ + θ) = cos(θ)cos(θ) - sin(θ)sin(θ)

4. Simplify:

   • cos(2θ) = cos²(θ) - sin²(θ)

5. To derive the other forms:

   • Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1.
   • Substitute to get:
     • cos(2θ) = 2cos²(θ) - 1
     • cos(2θ) = 1 - 2sin²(θ)

Common Pitfalls

• Forgetting to use the correct angle addition formulas can lead to mistakes in proving identities.
• Mixing up sine and cosine identities; ensure you use the right formulas for each function.

Step 3: Applying the Double Angle Identities

Once you’ve proven the identities, you can use them to simplify expressions or solve equations. Here’s how:

1. Simplifying Expressions:

   • Use the identities to rewrite expressions involving sin(2θ) or cos(2θ).
   • Example: Simplify sin(60°) using sin(2θ):
     • sin(60°) = sin(2 * 30°) = 2sin(30°)cos(30°)
     • Substitute known values: sin(30°) = 1/2, cos(30°) = √3/2.
     • Result: sin(60°) = 2 * (1/2) * (√3/2) = √3/2.

2. Solving Equations:

   • When faced with equations like sin(2x) = 0.5, use the double angle identity:
     • Set up the equation: 2sin(x)cos(x) = 0.5.
     • Solve for x using known values of sine and cosine.

Conclusion

Double Angle Identities are powerful tools in trigonometry that can simplify problems and enhance your understanding of angle relationships. Remember to practice proving and applying these identities to become proficient. As a next step, explore more complex trigonometric equations or delve into the applications of these identities in real-world scenarios, such as physics and engineering.