| |
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
|
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Exercises .....................................................................................................
22 |
| |
|
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Lesson
2 ..................................................................................................
25 |
| |
It is a duty of every triangle
to live by the laws of sine and cosine |
| |
Laws of cosine and sine |
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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.dads-lessons.com/trig/answers |
| |
Index ........................................................................................................
201
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