TR-39: Using Sum and Diff Identities (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial focuses on the Angle Sum and Difference Identities in trigonometry, as explained in Dennis F. Davis's video. These identities are essential for simplifying trigonometric expressions and solving equations involving sine, cosine, and tangent functions. Understanding these concepts can help you tackle problems in various math courses and standardized tests.

Step 1: Understanding Angle Sum Identities

Angle sum identities allow you to find the sine, cosine, or tangent of the sum of two angles. Here are the key identities:

• 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 Tips

• Use these identities to transform expressions involving sums of angles into simpler forms.
• When applying these identities, make sure to clearly define your angles A and B.

Step 2: Understanding Angle Difference Identities

Angle difference identities help in finding the sine, cosine, or tangent of the difference between two angles. The identities are:

• 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 Tips

• Similar to angle sum identities, these can simplify problems where angles are subtracted.
• Always remember the sign changes when dealing with the difference identities.

Step 3: Applying the Identities

To apply the angle sum and difference identities effectively, follow these steps:

1. Identify the Angles: Determine which angles you need to work with in your problem.
2. Choose the Right Identity: Decide whether to use angle sum or difference identities based on the given problem.
3. Substitute Values: Plug in the values of your angles into the respective identities.
4. Simplify: Carry out the calculations to simplify the expression or solve the equation.

Common Pitfalls

• Keep track of signs, especially when using difference identities.
• Ensure that the angles you are working with are in the same unit (degrees or radians) before applying the identities.

Step 4: Examples and Practice

Practicing with examples can solidify your understanding. Here are a couple of scenarios:

1. Example Problem: Find sin(75°) using the angle sum identity.

   • Set A = 45° and B = 30°.
   • Use the identity:
     • sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°).

2. Practice Problem: Solve for cos(15°) using the difference identity.

   • Set A = 45° and B = 30°.
   • Use the identity:
     • cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°).

Conclusion

Understanding and applying the Angle Sum and Difference Identities is crucial for solving trigonometric problems. Practice using these identities with different angles, and remember to be mindful of signs. As you become more familiar with these concepts, try tackling more complex trigonometric equations to further enhance your skills. For additional learning, consider reviewing related videos in the series, such as TR-38, for deeper insights.