Lecture Notes: Proving Trigonometric Identities

Overview

• Focus: Proving trigonometric identities.
• Builds on previous class knowledge.
• Use of formula sheet: Reciprocal, quotient, Pythagorean identities.
• Emphasis on a lot of practice and examples.
• Importance of independent practice.

What is an Identity?

• A mathematical expression true for all values of a variable.
• Verification: Plug in values to check if true.
• Proving: Show both sides of the expression are identical.

Cheat Sheet for Proving Identities

1. Operations Across Equal Sign: Not allowed; cannot subtract from both sides.
2. Simplify Independently: Work on both sides separately.
3. Complex to Simple: Start with the more complex side and simplify.
4. Use Known Identities: Apply known identities directly.
5. Change to Sines and Cosines: Beneficial to rewrite all terms in sines and/or cosines.
6. Factoring: Look for opportunities to factor expressions.
7. Fractions: Convert multiple fractions into a single fraction.
8. Conjugates: Occasionally multiply by conjugates.

Importance of Simplifying Identities

• Trigonometric functions are used in computing.
• Simplifying expressions can reduce computational load.

Example Problem: Verification & Proof

• Expression: ( \frac{1}{\cos x} - \cos x = \sin x \tan x )
• Verification: Check if true for ( x = \frac{\pi}{3} ).
• Proof:
  • Work on simplifying both sides separately.
  • Use known trigonometric identities to show equality.

Strategies for Solving Trigonometric Identities

• Use Cheat Sheet: Follow steps to simplify.
• Practice: Essential for mastering the process.

Additional Examples

• Work through multiple trigonometric identity proofs.
• Demonstrate step-by-step simplification.
• Show how to handle various types of identities and expressions.

Conclusion

• Practice Exercises: Page 314, questions 1 to 4, and 11b, 11c.
• Key Takeaway: Proving identities involves recognizing patterns, applying identities, and simplifying expressions.