Trigonometry

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Book 1 is pre-calculus trigonometry. We assume the student is relatively new to algebra and can do algebra step by step.

Many of the pages have closely related free/YouTube videos at the Khan Academy. This is by design. Many students find the video presentation helpful with learning mathematical material.

As with all three trigonometry books, we have a "For Enthusiasts" section, which is for the student who finds the normal content and pace too slow and too easy, and yet still needs exercises and practice with Book 1 trigonometry.

• Introduction

• Similar,Congruent,Isosceles, and Equilateral
• Angles of a triangle sum to 180 Degrees
• Exercise: Congruent Triangles
• Exercise: Chasing Angles
• The Pythagorean Theorem
• Proof: Pythagorean Theorem
• Exercise: A Puzzle Triangle
• Conventions
• Proof: Angles sum to 180
• Areas of Triangles
• Heron's Formula

• Plotting (Cos t, Sin t)
• Cosine and Sine
• Soh-Cah-Toa
• Sine Squared plus Cosine Squared
• Radians
• The Unit Circle
• Law of Sines
• Law of Cosines
• Solving Triangles Given ASA
• Solving Triangles Given SAS
• Worked Example: Area of a Roof
• Worked Example: Oil Tanker Arriving in Port
• * Exercises: Angles of Elevation and Depression
• The most Difficult Triangles to Solve

• Graphs of Sine, Cos and Tan
• Phase and Frequency
• Graph of Sine Squared
• Addition Formula for Sines
• Addition Formula for Cosines
• Double Angle Formulas
• * Waves in and out of Phase
• * Beat Frequencies
• * Worked Example: Ferris Wheel Problem
• * Worked Example: Simplifying Angles
• * Cosecant, Secant, Cotangent
• * Graphs of Cosec, Sec and Cot
• * Inverse Trigonometric Functions
• * Simplifying a sin(x) + b cos(x)

• * Remembering Trig Formulae
• * Trigonometric Formula Reference
• * Trigonometric Unit Circle and Graph Reference
• * Selected Angles Reference
• * Law of Tangents

• * Regular Polygons
• * What Tessellations are Possible with Regular Polygons?
• * Proof: Heron's Formula
• * Compass Bearings
• * Transformation of products into sums
• * The Distance from New York to Tokyo
• * Lissajous Figures
• * The CORDIC Algorithm
• * Nyquist Frequency
• * Pythagorean Triples
• * A Brief History of Trigonometry

• * About This Book (Book 1)
• * Khan Academy Videos
• * Common Student Errors (Book 1)

Book 2 is also pre-calculus trigonometry. However, the algebra moves at a brisker pace than in Book 1. The topics are not central to understanding trigonometry as it is usually taught in schools, now that a lot of former content has been dropped.

One rule of thumb of the topics in Book 2 is the union of the set of all topics in high-school contest related to trigonometry, applications, and the topics in the classical book Plane and Spherical Trigonometry by Palmer (link), subtracting any thoroughly discussed topics in Book 1, and excluding any topic that requires substantial use of calculus or the concept of limit (which should be done in Book 3).

The topics are useful, for example, for students interested in maths contests. In the enthusiasts section there are topics and exercises that are useful to students who will go on to do work with computer graphics.

Book 2 trigonometry deepens the understanding of the many relationships between triangles and circles. It also shows how to tackle some harder trigonometric function identities.

• Introduction

• * Geometric Definitions of Trig Functions
• * Thales Theorem: Quick construction of a right angle
• * Ellipses
• * Spirals

• * De Moivre's Formula
• * Verifying Trigonometric Identities
• * Solving Trigonometric Equations
• * Sum into Product (Exercises)
• * The sine of 18º
• * The sine of 15º
• * The transcendence of sine
• * The summation of sined or cosined arithmetic sequence
• * Other Trig Identities

• * Triangles on a Sphere
• * Application: The Distance from New York to Tokyo
• * Applications of spherical trigonometry (astronomy)

• * Related problems in practical use of trigonometry
• * Relation between sine x, x, tangent x for small x
• * Side opposite small angle given
• * Length of long sides given
• * Orthogonal projection
• * Bipolar (biangular) coordinates
• * Solving triangles by half-angle formulae
• * The distance and dip of the horizon
• * Cyclic Quadrilaterals and Ptolemy's Theorem
• * Area of a quadrilateral
• * Area of sector or segment
• * Widening of pavements on curves
• * Reflection and Refraction of a ray of light

• * The Circumcircle
• * The Incircle
• * The Excircles
• * Other Centres of a Triangle
• * Power of a Point
• * Ceva's Theorem
• * Orthocentric points
• * The Excentral Triangle
• * The Pedal Triangle
• * The Nine-point Circle
• * Brocard's Theorem
• * The Fermat Point
• * Reflections in a triangle
• * Morley's Triangle
• * Viviani's theorem
• * Napoleon's theorem
• * The Miquel Point
• * Philo's Line
• * Menelaus' theorem
• * Area bisectors
• * The Pivot theorem
• * The Seven circles theorem
• * The Simson line
• * Thebault's theorem

• * The solution of cubic equations

• * The Delaunay Triangulation

This section is for Book 2 pages where we don't yet know how they should fit in.

• * Some Little-known Trigonometric Functions

• * A Generalization of Pythagoras' Theorem
• * The Eyeball Theorem
• * Casey's Theorem

• * About This Book (Book 2)
• * Khan Academy Videos
• * Common Student Errors (Book 2)

These are pages that are on the way out.

• Trigonometric identities
• * Core concepts of trigonometry
• * Prerequisites and Basics
• Addition and Subtraction Theorems
• More About Addition Formulas - Some Content may need extracting to book 1.
• * Collection of Problems
• * Degrees Minutes and Seconds
• * Doing without Sine
• * Reflections in a line (possibly could be worked into an intro/review on matrices - book 3 intro chapters)

Book 3 uses and builds on calculus, complex numbers, matrices. We assume the student is relatively fluent with algebra. We will often combine simple steps to keep proofs/explanations short. Book 1 is a prerequisite, but book 2 isn't.

There are many computing related topics, particularly in the "for Enthusiasts" section.

• Introduction

• Some preliminary results
• Derivative of Sine
• Derivative of Cosine
• Derivative of Tangent
• Derivative of Inverse Functions
• Power Series for e^x
• Power Series for Cosine and Sine
• Numeric computation of Cosine and Sine
• e^iπ
• Cosh, Sinh and Tanh
• Functions of complex variables

• A Square Wave in Sines
• The Gibbs Overshoot
• The Plancherel Identity

• Eigenfunctions
• Representation Theory
• The Pontryagin Dual
• Wavelets

This section is for Book 3 pages where we don't yet know how they should fit in.

• Trig Substitutions
• Vectors in the Plane
• Vectors and Dot Products
• Trigonometric Form of the Complex Number
• Simple Harmonic Oscillator
• * Calculating π
• * Some theorems using vectors
• * Price's Theorem

• * Chebyshev Polynomials
• * Trigonometry done Rigorously
• * The Fast Fourier Transform
• * Erdős–Mordell inequality
• * Hilbert's third Problem
• * Wavelet Compression

• * About This Book (Book 3)
• * Common Student Errors (Book 3)

Lmov, Alsocal, Robinson0120,
Evil saltine, JEdwards, llg, Programmermatt, Douglas W. Mitchell

Also thanks to the many contributors to mathematical articles on Wikipedia from which some of the content has been lifted.