Trigonometry

For how to help with this book, click here. Do add new material and examples and make corrections. It all helps. Decide whether new material is book 1,2 or 3. We want book 1 to be ready to use in K12 education. Book 3, especially the for-enthusiasts parts can progress to post-graduate level - that's fine - as long as it's still recognisably trig. Have a look at About This Book, and even modify that, so we have a planned structure and so that it's easier for people to know where to add new content. All help is welcome. 350px|right The structure of this book is still being developed. The pages marked with a * are not currently accessed when reading the book in 'next page' order. Trigonometry /Print version/ Trigonometry The PDF Version of the book is not always up-to-date with the latest changes, so it is recommended to read the book online. Trigonometry Book 1 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 Enthusiasts25%" 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 100% Lengths, Angles and Areas in Triangles Similar,Congruent,Isosceles, and Equilateral 100% Angles of a triangle sum to 180 Degrees 100% Exercise: Congruent Triangles 100% Exercise: Chasing Angles 100% The Pythagorean Theorem 100% Proof: Pythagorean Theorem 100% Exercise: A Puzzle Triangle 100% Conventions 100% Proof: Angles sum to 180 100% Areas of Triangles 100% Heron's Formula 100% Trig Functions for Triangles Plotting (Cos t, Sin t) 75% Cosine and Sine 75% Soh-Cah-Toa 100% Sine Squared plus Cosine Squared 100% Radians 100% The Unit Circle 75% /Law of Sines/ 100% /Law of Cosines/ 100% Solving Triangles Given ASA 100% Solving Triangles Given SAS 100% Worked Example: Area of a Roof 25% Worked Example: Oil Tanker Arriving in Port 25% * Exercises: Angles of Elevation and Depression 50% The most Difficult Triangles to Solve 75% Trig Functions as Functions Graphs of Sine, Cos and Tan 75% Phase and Frequency 100% Graph of Sine Squared 100% Addition Formula for Sines 100% Addition Formula for Cosines 100% Double Angle Formulas 100% * Waves in and out of Phase 25% * Beat Frequencies 75% * Worked Example: Ferris Wheel Problem 100% * Worked Example: Simplifying Angles 75% * Cosecant, Secant, Cotangent 100% * Graphs of Cosec, Sec and Cot 75% * Inverse Trigonometric Functions 100% * Simplifying a sin(x) + b cos(x) 75% Trigonometry References * /Remembering Trig Formulae/ 100% * /Trigonometric Formula Reference/ 100% * /Trigonometric Unit Circle and Graph Reference/ 25% * /Selected Angles Reference/ 25% * /Law of Tangents/ 100% For Enthusiasts * Regular Polygons 25% * What Tessellations are Possible with Regular Polygons? 25% * Proof: Heron's Formula 75% * Compass Bearings 100% * Transformation of products into sums 25% * The Distance from New York to Tokyo50% * Lissajous Figures 50% * The CORDIC Algorithm 25% * Nyquist Frequency 0% * Pythagorean Triples 50% * A Brief History of Trigonometry 75% Teachers Notes * About This Book (Book 1) 75% * Khan Academy Videos 50% * Common Student Errors (Book 1) 0% Trigonometry Book 2 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 0% More Geometry * /Geometric Definitions of Trig Functions/ 25% * Thales Theorem: Quick construction of a right angle 25% * Ellipses 0% * Spirals 0% More on Trigonometric Identities * De Moivre's Formula * /Verifying Trigonometric Identities/ 25% * /Solving Trigonometric Equations/ 25% * /Sum into Product/ (Exercises) 25% * The sine of 18º 100% * The sine of 15º 75% * /The transcendence of sine/ 0% * The summation of sined or cosined arithmetic sequence 100% * Other Trig Identities 25% Going Spherical * Triangles on a Sphere75% * Application: The Distance from New York to Tokyo * Applications of spherical trigonometry (astronomy) Applications * 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 100% * The distance and dip of the horizon 100% * Cyclic Quadrilaterals and Ptolemy's Theorem 25% * Area of a quadrilateral 25% * Area of sector or segment * Widening of pavements on curves * Reflection and Refraction of a ray of light Circles, Points and Triangles Associated with a Triangle * The Circumcircle 25% * The Incircle 25% * The Excircles 75% * Other Centres of a Triangle 50% * Power of a Point 0% * Ceva's Theorem 50% * Orthocentric points 0% * The Excentral Triangle 50% * The Pedal Triangle 50% * The Nine-point Circle 50% * Brocard's Theorem 50% * The Fermat Point 75% * Reflections in a triangle 75% * Morley's Triangle 75% * Viviani's theorem 50% * Napoleon's theorem 50% * The Miquel Point 50% * Philo's Line 50% * Menelaus' theorem 75% * Area bisectors 50% * The Pivot theorem 50% * The Seven circles theorem 50% * The Simson line 50% * Thebault's theorem 50% Application to Algebra * The solution of cubic equations 75% Surveying Applications to Graphics * The Delaunay Triangulation 50% Where do These Belong? This section is for Book 2 pages where we don't yet know how they should fit in. * /Some Little-known Trigonometric Functions/ 100% for Enthusiasts * A Generalization of Pythagoras' Theorem 50% * The Eyeball Theorem 25% * Casey's Theorem 0% Teachers Notes * About This Book (Book 2) 75% * Khan Academy Videos 50% * Common Student Errors (Book 2) 0% Scrap Heap These are pages that are on the way out. /Trigonometric identities/ 25% * Core concepts of trigonometry 25% * /Prerequisites and Basics/ 25% /Addition and Subtraction Theorems/ 25% More About Addition Formulas 25% - Some Content may need extracting to book 1. * /Collection of Problems/ 25% * Degrees Minutes and Seconds 25% * Doing without Sine 25% * Reflections in a line 0% (possibly could be worked into an intro/review on matrices - book 3 intro chapters) Trigonometry Book 3 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 0% Calculus and complex numbers Some preliminary results 100% Derivative of Sine 100% Derivative of Cosine 100% Derivative of Tangent 100% Derivative of Inverse Functions 100% Power Series for ex 50% Power Series for Cosine and Sine 75% Numeric computation of Cosine and Sine eiπ 75% Cosh, Sinh and Tanh 75% Functions of complex variables 50% Basis Functions A Square Wave in Sines 0% The Gibbs Overshoot 0% The Plancherel Identity 0% Beyond the Fourier Transform Eigenfunctions 0% Representation Theory 0% The Pontryagin Dual 0% Wavelets 0% Where do These Belong? This section is for Book 3 pages where we don't yet know how they should fit in. Trig Substitutions 0% /Vectors in the Plane/ 25% /Vectors and Dot Products/ 25% /Trigonometric Form of the Complex Number/ 25% Simple Harmonic Oscillator 25% * Calculating π 25% * Some theorems using vectors 25% * Price's Theorem 25% for Enthusiasts * Chebyshev Polynomials 25% * Trigonometry done Rigorously 25% * The Fast Fourier Transform 0% * Erdős–Mordell inequality 25% * Hilbert's third Problem 50% * Wavelet Compression 0% Teachers Notes * About This Book (Book 3) 50% * Common Student Errors (Book 3) 0% Authors 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. K-12 mathematics Geometry T 50% __NOTOC__ da:Matematik A/Trigonometri de:Mathematik: Schulmathematik: Trigonometrie es:Trigonometría fr:CMC/3ème/Trigonométrie it:Trigonometria pt:Matemática elementar/Trigonometria