Download REFERENCES - mongolinternet.com

Fall 2004, Triginometry 1450-02, Week 6-7
Week 6
pp.121-148
Chapter 1. Trigonometry





What is the
TRIGONOMETRY o
Trigonometry=Angle+ Three sides + triangle + circle.
Trigonometry: Measurement of Triangles (derived form Greek language)
Tri means: Three (in English), drei (in German), tri (in Russian)  3 sides
n=0
Points
Numbers,
points
n=1
Line
Line coordinate,
Distance, absolute value
N=2
2 lines
Plane coordinate system
Angle, slope, circle, triangle
n=3
3 lines
Plane and Polar coordinate
Angle, Trigonometry
DEFINITION: Angle
 Angle: is determined by rotating a ray (half line).









 ,  ,  , A, B, C or A, B, C
Vertex of the angle: The endpoint of the ray. (Origin, central angle). O
Initial side of the angle: The starting position of the ray.
Terminal side of the angle: The position after rotation
Standard position: Plane coordinate system: Origin=vertex, initial side=positive x axis
Positive Angle: By Counter clockwise rotation. Quadrants 1,2,3,4.
Negative Angle: By clockwise rotation
Degree measure: equivalent to a rotation of 1 / 360 of a complete revolution about the vertex.
Radian: Measure of a central angle  .
Coterminal Angles: Angles  and  have the same initial and terminal sides.
0

One Revolution: (one full rotation), 360 or

Acute angle: 0    90   / 2

Right angle:


Complementary angles: Two positive angles













2
0
  90 0   / 2
0
0
Obtuse angle:  / 2  90    180  
 and  have sum is      / 2 ,  ,   0
Supplementary angles: Two positive angles  and  have sum is      ,  ,   0
0
Reference angles: Acute angle    90 by the terminal side of  and the horizontal axis x .
0
Circle: Center and Radius. A full angle of ac circle from its center equals to 360 or 2 radians.
Circumference: The perimeter of a circle. l  2r  2d
Central angle: Angle
AOC with endpoints A, C on a circle’s circumference and vertex O .
Arc of a circle: Any smooth curve joining two points of the circle by a central angle.
.
Length of Arc: s  r   .  / 2r   / 2
Chord of a circle: The line segment joining two points on a curve.
Circular segment: A portion (shaded region) of a circle whose upper boundary is arc and whose
lower boundary is a chord making a central angle    .
Circular sector: The entire wedge-shaped area.
Dis tan ce S
 (length of two points in real line)
Time
t
arc length s
 (length of the arc on the circle)
Linear speed: 
time
t
central angle 
 (central angle of the arc on the circle)
Angular speed: 
time
t
Speed: v 
Batmunkh.Ts
Math Graduate Student 10.07.2004
Page 1
Fall 2004, Triginometry 1450-02, Week 6-7
Some common angles in Degree measure and Radian measure
degree
00
30 0
45 0
60 0
90 0 120 0
Radian
0

6

4

3

2
135 0
150 0
3
4
5
6
2
3
180 0 270 0 360 0

3
2
2
Degree and Radian vs. Earth and Sun:
A full degree of an angle is 2  360 .
0
10  1day
Circular motion is most important motion. It is periodic.



The term Earth rotation refers to the spinning of the Earth on its axis with North Pole.
One rotation takes 24 hours and is called a mean solar day.
If we could see down at the Earth’s North Pole from space we would see that the direction of
rotation is counterclockwise. The clockwise direction is from the South Pole.



The Sun (center) is a star located at the center of our solar
system.
The orbit of the earth around the sun is called Earth revolution.
This circular motion takes 365 days (1 year) to complete one
cycle around the sun.
1 year (365) days to rotate
0
One year has 365 days like 360 . 4 seasons like 4 quadrants. Each season has 3 months. 12 months. 1
month has 30 days.
Batmunkh.Ts
Math Graduate Student 10.07.2004
Page 2
Fall 2004, Triginometry 1450-02, Week 6-7
Semicircle
  180 0
(radian and degree)
Arc and angle
Arrow, bow, chord
The Circle is Beauty of Shape. Maximum area for a given perimeter. Minimum perimeter for a given
area.
Similarity property.


l
L
l *  2


 unit circle r  1
2r 2 R
2
If you want to see molecules in your eyes, increase this until an apple. Then a medium apple goes
to the Earth.
Batmunkh.Ts
Math Graduate Student 10.07.2004
Page 3
Fall 2004, Triginometry 1450-02, Week 6-7
 rad  180 0
Convert to the radian measure
1rad 
Convert to the degrees measure
10 
180 0

180


180 0
 57.30
3.14
  
  
0

180 0
180 0

sin 2   cos 2   1
Sine, cosecant
y b opp
 
r c hyp
x a adj
cos    
r c hyp
sin 
y b
tan  
   slope
cos  x a
sin  
Cosine, secant
Tangent, cotangent
1
sin 
1
sec  
cos 
1
cot  
tan 
csc  
degree
00
30 0
45 0
60 0
90 0
120 0
135 0
150 0
Radian
0
sin 
1

3
3
2
1
2
2
3
3
2
1

2
1
3
Undef
3
4
2
2
2
2
1
5
6
1
2
cos

4
2
2
2
2

2
0

6
1
2
tan  
sin
cos 
0
3
2
1
3
1
0
 3
2
2 2
1
3
3 3
1
and




2
3
2 2
2
3 3
3
2
2
Sine
sin   cos   1
sin 


180 0 270 0 360 0

3
2
1
2
0
3
2
1
1
0
1
0
Undef
0
3
0

Domain

Range y  f (x)
Period
Odd, even functions
Batmunkh.Ts
Cosine
cos
(,)
(,)
 1  sin   1
sin   sin(   2n)
period 2
sin(  )   sin 
 1  cos  1
cos   cos(  2n)
period 2
cos(  )  cos 
Odd function
Even function
Math Graduate Student 10.07.2004
Tangent
tan  


sin 
cos 
 n
2
(,)
tan   tan(  n)
period 
tan(  )   tan 
Odd function
Page 4
Fall 2004, Triginometry 1450-02, Week 6-7
Sine function
Let  be an angle measured counterclockwise from the x -axis (initial side) along an arc of the unit
circle. Then sin   y is the vertical coordinate y of the arc endpoint. As a result of this definition, the
sine function is:
sin   sin(   2 )
 Periodic function with period 2 .

Odd function
sin(  )   sin 

Pythagorean Theorem:
sin 2   cos 2   1
Cosine function
Let  be an angle measured counterclockwise from the x -axis (initial side) along an arc of the unit
circle. Then cos   x is the horizontal coordinate x of the arc endpoint. As a result of this definition,
the sine function is:
cos   cos(  2 )
 Periodic function with period 2 .

Even function
cos(  )  cos 

Pythagorean Theorem:
sin 2   cos 2   1
Tangent function
The tangent function is defined by tan  
sin 
. Other notation is tan   tg
cos 
The word “tangent” also has an
important related meaning as a slope,
or tangent line or tangent plane.
Batmunkh.Ts
Math Graduate Student 10.07.2004
Page 5
Fall 2004, Triginometry 1450-02, Week 6-7
In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc
corresponding to the central angle
is denoted
. Similarly, the size of the central angle
subtended by this arc (i.e., the measure of the arc) is sometimes (e.g., Rhoad et al. 1984, p. 421) but not
always (e.g., Jurgensen 1963) denoted
.
The center of an arc is the center of the circle of which the arc is a part.
An arc whose endpoints lie on a diameter of a circle is called a semicircle.
For a circle of radius r, the arc length l subtended by a central angle
measured in radians, then the constant of proportionality is 1, i.e.,
Batmunkh.Ts
is proportional to
Math Graduate Student 10.07.2004
, and if
is
Page 6