Advanced Trigonometry

Triangles, circles and waves, and the identities that connect them

About this course

Trigonometry from similar triangles to the unit circle, periodic functions and their inverses, identities, general triangles, vectors and polar curves. No calculus is required.

What you will be able to do

  • Explain why trigonometric ratios depend only on angle, and work in radians
  • Graph sine, cosine and tangent as periodic functions and model periodic motion with them
  • Find every solution of a trigonometric equation using inverse functions and their branches
  • Prove identities, including compound, double-angle, half-angle and product formulas
  • Solve general triangles, including the ambiguous case, and apply them to bearings and vectors

What this course covers

  • One-to-one restriction and inverse branches
  • Principal value convention and notation
  • Inverse graph via axis reflection
  • Composition identities and their conditional validity
  • Principal range determines coordinate signs
  • Domain of composite inverse expressions
  • Two-check verification of principal results
  • Circle coordinates generate the sine and cosine graphs
  • Domain, range, and exact landmark values
  • Period as a whole-function property and the fundamental period
  • Even and odd symmetry from circle reflections and rotations
  • Quarter-turn relationship between sine and cosine graphs
  • Transforming a wave: amplitude, midline, period, and phase shift
  • Worked examples and closing synthesis
  • Similarity as the foundation of angle-dependent ratios
  • Defining sine, cosine, and tangent
  • Unit-hypotenuse picture and angle as input
  • Constructing special-angle triangles (30°, 45°, 60°)
  • Applying trigonometric ratios to find inaccessible heights
  • Logical chain: from similarity to trigonometric functions
  • Transfer: rotated diagrams and changed numbers
  • Triangle labeling and opposition convention
  • Derivation of the cosine rule
  • Two uses of the cosine rule
  • Derivation of the sine rule through area
  • Inverse sine ambiguity and when to use each rule
  • Information patterns and case selection
  • Solving triangles and multi-step verification
  • Trigonometric identities vs. conditional equations
  • The Pythagorean identity and its rearrangements
  • Domain tracking and the effect of simplification
  • Choosing and executing algebraic transformations
  • Conjugate factor technique and domain comparison
  • Auditing proofs: square roots, illegal cancellation, and circular reasoning
  • Reciprocal identity secant²θ − tan²θ = 1 and its applications
  • SSA Configuration and the Ambiguity Problem
  • Circle-and-Ray Construction for Counting Triangles
  • Acute Angle Case: Altitude and Side-Length Thresholds
  • Inverse Sine Candidates and the Angle-Sum Filter
  • Right and Obtuse Given Angle: One or No Triangle
  • Complete SSA Decision Procedure
  • Completeness, Existence vs Reconstruction, and Summary
  • Tangent as slope: definition, domain, and period
  • Vertical asymptotes and one-sided unbounded behavior
  • Secant and cosecant: reciprocal construction and range gap
  • Reciprocal vs. inverse function notation
  • Transforming the tangent graph
  • Removable holes vs. vertical asymptotes
  • Systematic procedure for sketching these graphs
  • Unit circle coordinate foundations
  • Rotating the two reference directions
  • Sine and cosine addition formulas
  • Subtraction formulas and exact special values
  • Tangent addition formula and its domain
  • Rotation as a 2×2 matrix
  • Exact coordinate calculations with rational ratios
  • Why one inverse value is incomplete
  • Complete angle families for sine, cosine, and tangent
  • Solving equations with a transformed argument
  • Factoring trigonometric equations into cases
  • Extraneous roots from squaring and denominator restrictions
  • Proof-of-completeness framework
  • Four habits for complete trigonometric solutions
  • The radian: arc length divided by radius
  • Radians versus degrees: conversion via a full turn
  • Arc length and sector area formulas
  • Arc versus chord: two different objects, same endpoints
  • Degree substitution error in radian formulas
  • Angular speed and linear speed in uniform rotation
  • Net displacement versus total travel in rotation
  • Notation distinctions: squared, doubled, and scaled outputs
  • Double-angle formulas for sine and cosine
  • Testing identities with special values
  • Power-reduction and half-angle formulas
  • Product-to-sum identities
  • Sum-to-product identities and wave envelopes
  • Solving equations with double-angle and product identities
  • Justifying the sinusoidal shape from uniform rotation
  • Sinusoidal model parameters and physical meanings
  • Period, frequency, and angular frequency relationships
  • Recovering parameters from extrema and event timing
  • Phase: starting position and initial direction
  • Limits of sparse constraints: alternative periodic models
  • Complete modeling methodology and habit summary
  • Normalization: from triangle ratios to unit-circle coordinates
  • Unit circle coordinate definition of sine and cosine
  • Reference triangles and quadrant signs
  • Coterminal angles and the information in an angle vs. an endpoint
  • Recovering a missing coordinate using the circle equation
  • Even and odd symmetry from horizontal-axis reflection
  • Systematic workflow for exact unit-circle problems
  • Bearings and signed displacement components
  • Vector addition of directed displacements
  • Dot product and vector projection
  • Relative velocity and heading vs. track
  • Existence condition for a direct crossing
  • Worked solutions to practice problems
  • Closing synthesis and geometric interpretation of projection
  • Polar coordinate system and conventions
  • Non-unique polar addresses and equivalence rules
  • Converting between polar and Cartesian coordinates
  • Polar circle r = 4 cos θ and its Cartesian identity
  • Cardioid r = 2(1 + cos θ) and non-negative radius tracing
  • Rose curve r = 2 cos 3θ and petal counting via sign intervals
  • Intersection of polar curves and the separate pole check
  • Cosine wave model: parameters and units
  • Model assumptions and their role
  • Non-uniqueness of a model fitted to sparse data
  • Discriminating between rival models with a withheld reading
  • Solving a cosine inequality for access intervals
  • Guaranteed vs possible access under a uniform error bound
  • Defending a model: assumptions, tests, limits, and course synthesis

Syllabus

  1. Ratios, Radians and the Unit Circle

    1. L1 · Why Triangle Ratios Depend Only on Angle
    2. L2 · Radians and the Geometry of Angle
    3. L3 · From Triangles to the Unit Circle

  2. Trigonometric Functions and Periodic Models

    1. L4 · How a Circle Draws a Wave
    2. L5 · Tangent, Reciprocals and Asymptotes
    3. L6 · Modelling Periodic Motion

  3. Inverses and Equations

    1. L7 · Inverse Functions and Their Branches
    2. L8 · Finding Every Trigonometric Solution

  4. Identities and Rotations

    1. L9 · Proving Trigonometric Identities
    2. L10 · Combining Rotations
    3. L11 · Double Angles, Half Angles and Products

  5. Triangles, Vectors and Polar Curves

    1. L12 · Solving General Triangles
    2. L13 · The Ambiguous Triangle
    3. L14 · Bearings, Vectors and Projections
    4. L15 · Polar Coordinates and Curves
    5. L16 · Building and Defending a Model