# Trigonometry

- 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.

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. |

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[edit]

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

### Lengths, Angles and Areas in Triangles[edit]

- 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

### Trig Functions for Triangles[edit]

- 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

### Trig Functions as Functions[edit]

- 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)

### Trigonometry References[edit]

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

### For Enthusiasts[edit]

- * 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

## Teachers Notes[edit]

## Trigonometry Book 2[edit]

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.

### More Geometry[edit]

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

### More on Trigonometric Identities[edit]

- * 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

### Going Spherical[edit]

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

### Applications[edit]

- * 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

#### Circles, Points and Triangles Associated with a Triangle[edit]

- * 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

#### Application to Algebra[edit]

#### Surveying[edit]

#### Applications to Graphics[edit]

### Where do These Belong?[edit]

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

### for Enthusiasts[edit]

## Teachers Notes[edit]

### Scrap Heap[edit]

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)

## Trigonometry Book 3[edit]

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.

### Calculus and complex numbers[edit]

- 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

### Basis Functions[edit]

### Beyond the Fourier Transform[edit]

### Where do These Belong?[edit]

*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

### for Enthusiasts[edit]

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

## Teachers Notes[edit]

## Authors[edit]

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.