Overview
This lecture covers multiple and sub-multiple angle formulas in trigonometry, key manipulations, and their applications in solving various types of problems, including special angles and triple-angle identities.
Multiple and Sub-Multiple Angle Formulas
- sin 2θ = 2 sin θ cos θ
- sin 2θ = (2 tan θ) / (1 + tan² θ)
- sin θ = 2 sin(θ/2) cos(θ/2)
- cos 2θ = cos² θ – sin² θ
- cos 2θ = 2 cos² θ – 1
- cos 2θ = 1 – 2 sin² θ
- cos 2θ = (1 – tan² θ) / (1 + tan² θ)
- tan 2θ = (2 tan θ) / (1 – tan² θ)
Gold Formulas
- 2 cos² θ = 1 + cos 2θ
- 2 sin² θ = 1 – cos 2θ
Triple Angle Formulas
- sin 3θ = 3 sin θ – 4 sin³ θ
- cos 3θ = 4 cos³ θ – 3 cos θ
- tan 3θ = (3 tan θ – tan³ θ) / (1 – 3 tan² θ)
Problem-Solving Techniques
- For proofs, rewrite all terms in sine and cosine for easy manipulation.
- To solve identities or simplify, use formulas to convert higher angles to single θ or θ/2.
- For expressions like sin θ/(1 + cos θ), substitute using half-angle formulas for both numerator and denominator.
- For values like sin 22.5°, utilize double-angle or half-angle formulas and plug in known standard angles.
- Special values: sin 18° = (√5 – 1) / 4.
Application Examples
- Use trigonometric identities to simplify or prove expressions as shown in the sample problems (e.g. tan 2a calculation, half-angle reductions).
- Manipulate expressions by multiplying/dividing by strategic values (e.g. 2 or known trigonometric constants) to facilitate simplification.
Key Terms & Definitions
- Multiple Angle — An angle that is a multiple of a given angle (e.g. 2θ, 3θ).
- Sub-Multiple Angle — An angle that is a fraction of another angle (e.g. θ/2).
- Gold Formulas — Key double-angle identities for sin² θ and cos² θ.
- Half-Angle Formula — Expresses functions of θ in terms of θ/2.
Action Items / Next Steps
- Review and memorize all core formulas listed above.
- Solve practice questions, especially those involving special angles (e.g. 15°, 22.5°, 18°).
- Watch the provided four-month math planner video and set up a custom weekly/daily study schedule.
- Prepare for the next practice session at 10 o'clock featuring function problems.
- Optional: Suggest topics or needs (like Olympiad prep) in the comments.