Overview
This lecture covers strategies and key identities for simplifying trigonometric expressions, emphasizing the use of reciprocal and Pythagorean identities.
Reciprocal and Quotient Identities
- Secant (sec θ) = 1 / cos θ
- Cosecant (csc θ) = 1 / sin θ
- Tangent (tan θ) = sin θ / cos θ
- Cotangent (cot θ) = cos θ / sin θ
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Simplification Examples
- Replacing sec θ with 1 / cos θ and canceling terms can reduce expressions, e.g., sec θ × sin θ / cos θ = sin θ.
- tan²θ + sin²θ + cos²θ simplifies to sec²θ using Pythagorean identities.
- sin²θ × csc θ × sec θ simplifies to tan θ by substitution and cancellation.
- cot θ × tan θ + cot²θ simplifies to csc²θ through distribution and identity conversion.
- sec²θ × (1 − sin²θ) simplifies to 1 by substituting (1 − sin²θ) with cos²θ and canceling terms.
- cos θ + sin θ × tan θ becomes sec θ by rewriting tan θ as sin θ / cos θ, finding a common denominator, and applying the Pythagorean identity.
- [sec θ × (sin θ × cot θ + sin θ × tan θ)] simplifies to 1 using identities, distribution, and reducing complex fractions.
Key Terms & Definitions
- Reciprocal identities — Express trig functions as reciprocals (e.g., sec θ = 1 / cos θ).
- Quotient identities — Express tangent and cotangent as ratios of sine and cosine.
- Pythagorean identities — Fundamental relationships among trig functions (e.g., sin²θ + cos²θ = 1).
- Simplification — Reducing trig expressions using algebraic manipulation and identities.
Action Items / Next Steps
- Memorize reciprocal, quotient, and Pythagorean identities.
- Practice simplifying trigonometric expressions using these identities.
- Prepare for exams by working through similar example problems.