Overview
This lecture introduces key properties of trigonometric functions, focusing on periodicity, fundamental identities, and function symmetry (even/odd). It includes important examples and emphasizes essential identities to memorize.
Periodicity of Trigonometric Functions
- A function is periodic if F(θ + P) = F(θ) for some nonzero P, called the period.
- Sine, cosine, cosecant, and secant have period 2π; tangent and cotangent have period π.
- For sine and cosine: sin(θ + 2πk) = sin(θ), cos(θ + 2πk) = cos(θ) for any integer k.
Finding Exact Values Using Periodicity
- Use the unit circle and periodicity to find sine, cosine, and tangent for large angles.
- Example: sin(17π/4) = √2/2; cos(5π) = –1; tan(5π/4) = 1.
Fundamental Trigonometric Identities
- Pythagorean Identity: sin²θ + cos²θ = 1.
- Tangent/Secant Identity: 1 + tan²θ = sec²θ ("I tan in a second").
- Cotangent/Cosecant Identity: 1 + cot²θ = csc²θ ("I cotan and a cosecant").
- These three identities are essential and should be memorized.
Notation for Trig Functions
- sin²θ means (sin θ)²; similar notation applies for cosine and tangent.
Using Identities: Example Problems
- To simplify or evaluate trig expressions, rewrite using identities.
- Example: sin²(π/12) + 1/sec²(π/12) = 1, by Pythagorean identity.
Determining All Trig Functions from Sine or Cosine
- If sin θ = 1/3 and cos θ < 0, use the Pythagorean identity and signs to find all trig function values.
- Use coordinates or the unit circle and solve for the unknowns using identities.
Even and Odd Functions
- Even function: f(–θ) = f(θ), e.g., cosine and secant.
- Odd function: f(–θ) = –f(θ), e.g., sine, tangent, cosecant, cotangent.
- Evenness/oddness can simplify evaluation and integration of trig functions.
Applying Even/Odd Properties: Examples
- sin(–45°) = –sin(45°) = –√2/2 (sine is odd).
- cos(–π) = cos(π) = –1 (cosine is even).
- tan(–37π/4) = –tan(37π/4) = –1 (tangent is odd).
Key Terms & Definitions
- Periodic function — A function that repeats values in regular intervals.
- Period — The interval length P for which f(θ + P) = f(θ).
- Pythagorean identity — An equation relating sin²θ and cos²θ to 1.
- Even function — f(–θ) = f(θ); symmetric about the y-axis.
- Odd function — f(–θ) = –f(θ); symmetric about the origin.
Action Items / Next Steps
- Memorize the three key trigonometric identities.
- Complete homework problems involving periodicity, identities, and even/odd function properties.
- Read the next section on graphing trigonometric functions.