Rucete ✏ AP Precalculus In a Nutshell
8. Trigonometric Functions
This chapter introduces trigonometric functions and how they model periodic behavior. You’ll learn about the sine, cosine, and tangent functions, graphing, transformations, inverse functions, and solving trig equations—all crucial tools for modeling oscillating behavior in math and the real world.
- Periodic Phenomena and the Unit Circle
- Periodic functions repeat values over regular intervals (called a cycle).
- Examples: daylight hours, moon phases, temperature cycles
- Period = smallest positive value k where f(x + k) = f(x)
- Unit circle (radius = 1) used to define trig functions
- Radian measure: π radians = 180°, full circle = 2π radians
- Sine, cosine, tangent defined via point (x, y) on unit circle:
- sin(θ) = y, cos(θ) = x, tan(θ) = y/x
- Sine and Cosine Functions
- Sine and cosine repeat every 2π ⇒ period = 2π
- Range = [−1, 1]; domain = all real numbers
- Use unit circle and key angles: π/6, π/4, π/3
- Reflections across axes help determine quadrant values
- Sine graph: starts at 0 → 1 → 0 → −1 → 0
- Cosine graph: starts at 1 → 0 → −1 → 0 → 1
- Sine is an odd function: sin(−x) = −sin(x)
- Cosine is an even function: cos(−x) = cos(x)
- Sinusoidal Transformations
- General form: f(θ) = a sin(b(θ + c)) + d
- a = amplitude
- b = frequency → period = 2π/b
- c = phase shift
- d = vertical shift
- Graphing steps:
- Find amplitude and period
- Scale x-axis by ¼ period
- Apply shifts
- Reflect if amplitude is negative
- Cosine graph is sine shifted left π/2
- Tangent and Reciprocal Functions
- tan(θ) = sin(θ)/cos(θ); undefined at cos(θ) = 0
- Period = π; x-intercepts at multiples of π
- No amplitude; increases between asymptotes
- sec(θ) = 1/cos(θ), csc(θ) = 1/sin(θ), cot(θ) = 1/tan(θ)
- sec and csc form “U” shapes; undefined where original is 0
- cotangent decreases; period = π
- Inverse Trigonometric Functions
- Restricted domains allow inverses:
- sin⁻¹(x): [−1, 1] → [−π/2, π/2]
- cos⁻¹(x): [−1, 1] → [0, π]
- tan⁻¹(x): ℝ → (−π/2, π/2)
- Evaluate with unit circle or calculator
- Also written as arcsin, arccos, arctan
- Graphs are reflections over y = x
- Trig Equations and Inequalities
- Solve by isolating trig expression and using inverse trig
- Use reference angles and unit circle to find all solutions
- Add coterminal angles for unrestricted domains
- Inequalities use graphs or unit circle regions
- Modeling with Trigonometric Functions
- Used for periodic data: temperature, tides, sound, etc.
- Model: y = a sin b(x − c) + d
- a = amplitude
- b = 2π/period
- c = horizontal shift
- d = midline
- Estimate parameters from data or regression tools
- Graphing Summary
- All trig graphs can be reflected, shifted, stretched
- Amplitude = (max − min) / 2
- Midline = (max + min) / 2
- Period = 2π/b (sine/cosine), π/b (tangent/cotangent)
- Phase shift = −c
- Plot using key points and transformations
- Trig Identities and Simplification
- Fundamental identity: sin²θ + cos²θ = 1
- Derived identities:
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
- Used to simplify and solve trig equations
- Inverse trig functions can be verified via triangle definitions
In a Nutshell
Trigonometric functions are powerful tools for modeling periodic behavior. From sine and cosine waves to solving equations and transforming graphs, these functions give us a way to represent oscillations, angles, and cycles. Mastering the unit circle, trig identities, and transformations allows you to connect geometry, algebra, and real-world patterns with confidence.