Dad's Lessons: Trigonometry. "Sine of acute angle in a right triangle is the ratio of the opposite side to the hypotenuse. Let's examine the properties of sine."
Stop! Stop!! What a strange definition? Where did it come from? Why do we need to study some sine? Is it only to pass an exam?
Dear reader! Do similar questions arise in your mind when you first face such definitions? Does an inner feeling of protest and some premature disgust appear towards these "sines", with which you are deliberately tortured? If so, then this is the book for you. Our goal is to completely destroy such thoughts.
To achieve this goal, avoid formal style and dry language, and present material in a scientifically strong, yet lively and intuitive manner, we developed this book in the form of conversations between father - an expert in the field, and his daughter and son.
The children are not familiar with trigonometry at all and do not have any special mathematical abilities. However, they do have one very valuable trait: they are curious.
We are deeply convinced that this is the most important quality you need to have to become a master of any subject.
The conversations between the father and his kids are written in a simple and causal language. You can consider these conversations as some sort of scientific "home" seminar, where the children argue, make mistakes, and sometimes unknowingly, make serious scientific discoveries. Therefore, by reading this textbook you will not only acquire knowledge about trigonometry, but will also see how science is "made". During the course of learning, ask yourself this question: Would I have been able to come up with a certain result myself, or is it something completely incredible?
To emphasize the main initial idea of trigonometry about similarity of triangles, we decided to put on the cover page Curry's puzzle modified by Martin Gardner and presented with his kind permission,
We hope that once you've read this book, you will say: "You know, trigonometry isn't so disgusting after all. Actually, it is even beautiful."
· The main goal of this book is to present the material in a scientifically strong, yet lively and enjoyable form.
· Special care is made for enlightenment of motivations of ideas, principles and the paths to their discovery.
· This books caters to a wide range of audiences: from High School and College students to teachers and college instructors of different levels, as well as for Self-Teaching
Above are extracts from the book from the book Dad's Lessons: Trigonometry (with permission of the publishers).
There are no teacher's guide or workbook. Answers are provided at the back of the book. Full answers are provided at the authors' website.
Dad's Lessons: Trigonometry
- Lesson 1.....Protect your nose, study trigonometry!
Definition of trigonometric functions
- Problem of calculating the height of a tree
- Properties of similar triangles
- Finding the height of a tree and the definition of tangent
- Calculating distance on a rough terrain and the definition of sine
- Definitions of cosine, cotangent, secant and cosecant
Exercises
- Lesson 2.....It is the duty of every triangle to live by the laws of sine and cosine
Laws of cosine and sine
- Problem of finding a side of a triangle using two other sides and the angle in between them
- Generalization of the Pythagorean Theorems (law of cosines)
- Problem of finding a side of a triangle using another side and two angles
- The proportion of sides and sines of angles (law of sines)
- Solving triangles
Exercises
- Lesson 3.....Angles - acute, properties - cute
Simplest properties of trigonometric functions for acute angles
- Connection of secant, cosecant and cotangent with cosine, sine and tangent
- Expression of tangent in terms of sine and cosine
- Main identity for sine and cosine
- Values of trigonometric functions for the angles of 30o, 45o, and 60o
- Reduction formulas
Exercises
- Lesson 4.....Let's give each angle a trigonometric function!
General definition of trigonometric functions
- Definition of trigonometric functions for angles of 0o and 90o
- Concept of negative angles
- Angles on a unit circle in a coordinate system
- Definition of sine, cosine and tangent for any angle
Exercises
- Lesson 5.....Obtuse angles follow next, still the properties aren't complex
Simplest properties of general trigonometric functions
- The main identity
- Ranges of sine and cosine
- Reduction formulas
- Even and odd properties
- The "head" rule to memorize reduction formulas
- Values of trigonometric functions for the angles of 180o and 270o
- Periodic properties
Exercises
- Lesson 6.....The queen formula
Formula for the cosine of the difference of two angles
- Expression of the length of a segment through the coordinates of the end points
- Derivation of the formula for the cosine of difference of two angles
- Calculating trigonometric functions for the angle of 15o
Exercises
- Lesson 7.....The queen move
Main formulas for trigonometric functions
- Formula for cosine of sum of two angles
- Formula for sine of sum and difference of two angles
- Formulas for multiplication of sines and cosines
- Formulas for sum and difference of sines and cosines
- Formulas for double and half angles
- Calculation of cosecant for the angle of 1995o
- Calculating trigonometric functions for the angle of 18o
Exercises
- Lesson 8.....Alien measure of angles or the mystery of agent 0.017
Radian measure of angles
- Definition of a radian
- Expression of the length of a circle arc through the central angle and the radius
- Relation between degrees and radians
Exercises
- Lesson 9.....Through the sine waves to the vastness of the universe
The graph of sine
- Representation of angles as points on a coordinate axis
- Construction of the graph of sine for acute angles
- Construction of the graph of sine for obtuse angles
- Construction of the graph of sine for all angles
Exercises
- Lesson 10.....Crashing the sine wave against the cosine, or something about the splashes of tangent
Graph of cosine and tangent
- The rule for construction of a graph of a "shifted" function
- Construction of the graph of cosine
- Construction of the graph of tangent for angles from 0 to ð
- Construction of the graph of tangent for all angles
Exercises
- Lesson 11.....Triangle problems and trigonometric equations
Solving of the simplest trigonometric equations
- Solving the equation sin x = A for special values of A = 1, -1, 0, ½
- Solving the equation cos x = A for special values of A = 1, -1, 0, ½
Exercises
- Lesson 12.....Equations are good, but inverse functions are better!
The function inverse to cosine
- Analysis of the function
as inverse to y = x2
- The rule for construction of the graph of an inverse function
- Role of monotonic regions
- Definition of the function y = arccos x as an inverse to cosine
- Graph of the function y = arccos x
- Solving the equation cos x = A for any A
Exercises
- Lesson 13.....Let's convert sine and tangent to new functions!
Function inverse to sine and tangent
- Definition of the function y = arcsin x as an inverse to sine
- Graph of the function y = arcsin x
- Solving the equation sin x = A for any A
- Definition of the function y = arctan x as an inverse to tangent
- Graph of the function y = arctan x
- Solving the equation tan x = A for any A
- The rule to memorize range of inverse trigonometric functions
Exercises
- Lesson 14.....We'll solve any problem!
Additional properties and problems
- Analysis of expressions sin (arcsin x), cos (arccos x) and arcsin (sin á)
- Calculation of arcsin (sin 6ð/7)
- Calculation of sin(arccos x) and cos(arcsin x)
- Calculation of cos(arctan x)
- Calculation of arcsin x + arccos x
- Analysis of "even" properties of inverse trigonometric functions
- Solving the equation sin á = cos á
Sample
Lesson 1
Lesson 2
Lesson 8