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Lesson 1 .................................................................................................. 13
 
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 ..................................................................................................... 22
   
  Lesson 2 .................................................................................................. 25
  It is a 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 Theorem (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 ..................................................................................................... 34
   
  Lesson 3 .................................................................................................. 37
  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 30°, 45°, and 60°
Reduction formulas
  Exercises ..................................................................................................... 44
   
  Lesson 4 .................................................................................................. 47
  Let’s give each angle a trigonometric function!
  General definition of trigonometric functions
  Definition of trigonometric functions for angles of 0° and 90°
Concept of negative angles
Angles on a unit circle in a coordinate system
Definition of sine, cosine and tangent for any angle
  Exercises ..................................................................................................... 59
   
  Lesson 5 .................................................................................................. 61
  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 180° and 270°
Periodic properties
  Exercises ..................................................................................................... 75
   
  Lesson 6 .................................................................................................. 77
  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 15°
  Exercises ..................................................................................................... 84
   
  Lesson 7 .................................................................................................. 85
  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 1995°
Calculation of sine for the angle of 18°
  Exercises ..................................................................................................... 95
   
  Lesson 8 .................................................................................................. 97
  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 ..................................................................................................... 107
   
  Lesson 9 .................................................................................................. 109
  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 ..................................................................................................... 118
   
  Lesson 10 .................................................................................................. 119
  Crashing the sine wave against the cosine, or something about the splashes of tangent
  Graphs 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
Construction of the graph of tangent for all angles
  Exercises ..................................................................................................... 127
   
  Lesson 11 .................................................................................................. 129
  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 .....................................................................................................139
   
  Lesson 12 .................................................................................................. 141
  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 .....................................................................................................153
   
  Lesson 13 .................................................................................................. 155
  Let’s convert sine and tangent to new functions!
  Functions 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 ..................................................................................................... 163
   
  Lesson 14 .................................................................................................. 165
  We’ll solve any problem!
  Additional properties and problems
  Analysis of expressions sin(arcsin x), cos(arccos x) and arcsin(sin)
Calculation of arcsin(sin 6/7)
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 + cos
Solving the equation sin = sin 2
Solving the equation sin 5 = sin 7
Solving the equation sin + sin 2 + sin 3 = 0
Solving the equation a.sin + b.cos = c for any constants a, b, c
  Exercises ..................................................................................................... 178
   
  Summary of Results ................................................................................. 181
  Answers to Exercises ................................................................................ 189
  For full answers, please see http://www.dadslessons.com/trig/answers
 

Index ........................................................................................................ 201

 

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