Lecture on Trigonometry Functions

Jul 11, 2024

Trigonometry Functions Lecture

Introduction

  • Welcome and request for comments for motivation.
  • Previous lecture covered Algebraic functions.
  • Current focus: Trigonometry functions, specifically for 12th grade.
  • Importance of understanding and memorizing formulas.

Key Formulas and Tricks

Basic Trigonometric Formulas

  • Sine and Cosine: When different, use the sine function twice. When similar, use the cosine function twice.
  • Order: If sine comes first, the result is positive; if sine comes second, the result is negative.

Half-Angle Formulas

  • For $\cos \theta$, use formulas like \cos(2\theta) = 1 - 2sin²(\theta).
  • Key formulas: $\sin^2x = 1 - \cos(2x)$, $\cos^2x = 1 - 2sin²(2x)$.

Application of Formulas

  • Apply sine and cosine rules based on angle relationships.
  • Always verify angle sizes before applying formulas.

Derivatives and Integrals

  • Basic form: $\frac{d}{dx}[a^x \cos b x + a^x sin bx]$.
  • Understand adding and subtracting angles in trigonometric functions to simplify expressions.

Example Problems

Problem 1

  • Given: $\cos(6x) - \cos(2x)$
  • Approach: \cos formula application, simplify using trigonometric identities.
  • Solution: Break down to basic trigonometric forms and simplify.

Problem 2

  • Given: $\sin^2x + \cos^2x$
  • Approach: Utilize known identities like $\sin^2x = 1 - \cos(2x)$. Adjust angles accordingly and solve.
  • Solution: Apply double angle formulas and simplify.

Problem 3

  • Given: Convert expressions involving $\sin(2x)$ and $\cos(2x)$.
  • Approach: Rewrite using basic and double angle identities.
  • Solution: Simplify step-by-step, balancing terms as necessary.

Key Takeaways

  • Consistent application of sine and cosine rules simplifies solving complex trigonometric problems.
  • Memorizing basic and derived identities is critical for swift problem-solving in exams.
  • Practical understanding and application of these formulas can aid in university-level math.

Tips for Success

  • Regular practice and review of formulas and their applications.
  • Break down complex problems into simpler parts using known identities.
  • Engage with study groups or forums to share insights and problem-solving techniques.

Conclusion

  • Encouragement to practice more problems from this topic.
  • Request for feedback and sharing the content for motivation and help.
  • Acknowledgement of viewer support and sign-off.