Essential Trigonometry Concepts for IB Math

Aug 24, 2024

Trigonometry Recap for IB Math AASL

Overview

  • This is the second part of a trigonometry recap.
  • Focuses on trigonometric functions and trig equations.
  • Not exhaustive but a good overview.
  • Emphasizes multiple ways to solve trig problems.

Sinusoidal Functions

  • Functions: Sine (sin x) and Cosine (cos x) are sinusoids.
  • Period: Both have a default period of 2π.
    • Period: Width of one full cycle.
  • Amplitude: Height from max to midline; default is 1 from -1 to 1.
  • Midline (Principal Axis): Default is on y = 0.
  • Graph Characteristics:
    • Sine starts in the middle and moves up.
    • Cosine starts high and moves down.

Modeling with Sin and Cosine

  • Equation Parameters:
    • A: Amplitude (vertical stretch).
    • B: Period adjustment (horizontal stretch).
    • C: Horizontal shift.
    • D: Vertical shift (midline location).
  • Example:
    • 3 cos(5x): Amplitude = 3, Period = 2π/5.
    • -5 sin(x/4): Amplitude = 5, Period = 8π.

Trigonometric Equations

  • Graphical Solutions:
    • Use GDC for solutions in paper 2.
  • Analytic Techniques:
    • Isolate trig ratio.
    • Find reference angles.
    • Determine quadrants.
    • Write solutions.
  • Unit Circle: Use for reference angles and quadrant determination.

Solving Trig Equations

  • Steps:
    1. Isolate the trig function.
    2. Find the reference angle.
    3. Determine possible quadrants for solution.
    4. Adjust for any transformations (e.g., 2x).
  • Periodic Solutions: Solutions recur based on the function's period.

Special Cases

  • Quadratic Trig Equations: Factor like a quadratic equation.
  • Sneaky Equations: Convert to tangent if sine and cosine are present.
  • Messy Equations: Use identities to simplify mixed trig functions into a single type.

Specific Problem Types

  • Cosine Graph:
    • Example: Find the equation given a graph.
    • Steps: Identify midline, amplitude, period, shift.
  • Solving Equations:
    • Use unit circle or graph for reference angles.
    • Consider periodic nature of solutions.

Tools and Methods

  • Graphical Methods: Verify solutions using graphing calculator.
  • Remembering Values: Use tables, hand tricks, or special triangles for trig values.
  • Conversion: Use radians and degrees appropriately.

Key Takeaways

  • Sinusoidal modeling applies to many real-life periodic phenomena.
  • Understanding transformations and periodic nature is crucial in solving trig equations.
  • Practice with a variety of problems to master concept application.