Trigonometry 101

TRIGONOMETRY MATH 102

2ndSEM/SY2014-2015

Consultation Time: 2:30 4:30 PM

Main Book:

Algebra and Trigonometry by Loius Liethold

Reference Book:

UM Core Values

Excellence

Honesty and Integrity

Teamwork

Innovation

Course Description

Trigonometric functions; identities and

equations; solutions of triangles; law of sines;

law of cosines; inverse trigonometric

functions; spherical trigonometry

5

Course Objectives

After completing this course, the student must be able to:

1. Define angles and how they are measured;

2. Define and evaluate each of the six trigonometric functions;

3. Prove trigonometric functions;

4. Define and evaluate inverse trigonometric functions;

5. Solve trigonometric equations;

6. Solve problems involving right triangles using trigonometric

function definitions for acute angles; and

7. Solve problems involving oblique triangles by the use of the sine

and cosine laws.

6

Course Outline

1. Trigonometric Functions

1.1. Angles and Measurement

1.2. Trigonometric Functions of Angles

1.3. Trigonometric Function Values

1.4. The Sine and Cosine of Real Numbers

1.5. Graphs of the Sine and Cosine and Other Sine Waves

1.6. Solutions of Right Triangle

7

Course Outline

2. Analytic Trigonometry

2.1. The Eight Fundamental Identities

2.2. Proving Trigonometric Identities

2.3. Sum and Difference Identities

2.4. Double-Measure and Half-Measure Identities

2.5. Inverse Trigonometric Functions

2.6. Trigonometric Equations

2.7. Identities for the Product, Sum, and Difference of

Sine and Cosine

8

Course Outline

3. Application of Trigonometry

3.1. The Law of Sines

3.2. The Law of Cosines

4. Spherical Trigonometry

4.1. Fundamental Formulas

4.2. Spherical Triangles

9

TRIGONOMETRY

A branch of Geometry

Developed from a need to compute angles

and distances

Until about the 16th century, trigonometry

was chiefly concerned with computing the

numerical values of the missing parts of a

triangle when the values of other parts were

given.

10

Branches of TRIGONOMETRY

Plane

Problems involving angles and distances in one

plane/flat surfaces

Spherical-

Applications to similar problems in more than

one plane of three-dimensional space

curved surfaces

11

Plane vs Spherical

The sum of the angles of a spherical triangle is always greater than 180

In the planar triangle the angles always sum to exactly 180.

12

Application of Trigonometry

Carpentry

Mechanics

Machine work

Astronomy

Land survey and

measurement

Map making,

Artillery range

finding.

And others

13

Greek Word (origin)

14

History of TRIGONOMETRY

History of Trigonometry

Several ancient civilizationsin particular, the Egyptian, Babylonian, Hindu, and Chinesepossessed a considerable knowledge of practical geometry, including some concepts that were a prelude to trigonometry.

15

16

The Rhind papyrus, an Egyptian collection of 84 problems in arithmetic, algebra, and geometry dating from about 1800 BC, contains five problems dealing with the seked.


17

For example, problem 56 asks: If a pyramid is 250 cubits high and the side of its base is 360 cubits long, what is its seked? The solution is given as 51/25 palms per cubit; and since one cubit equals 7 palms, this fraction is equivalent to the pure ratio 18/25.


18


19

Trigonometry began with the

Greeks.

Hipparchus (c. 190120 BC) was

the first to construct a table of

values for a trigonometric function. Astronomer

founder of trigonometry


20

He considered every triangleplanar or sphericalas being inscribed in a circle, so that each side becomes a chord (that is, a straight line that connects two points on a curve or surface.


21

To compute the various parts of the triangle, one has to find the length of each chord as a function of the central angle that subtends itor, equivalently, the length of a chord as a function of the corresponding arc width.


22

The first major ancient work on trigonometry to reach Europe intact after the Dark Ages was the Almagest by Ptolemy (c. AD 100170).

He lived in Alexandria, the intellectual centre of the Hellenistic world, but little else is known about him.


23


Chapters 10 and 11 of the first book of the Almagest deal with the construction of a table of chords, in which the length of a chord in a circle is given as a function of the central angle that subtends it, for angles ranging from 0 to 180 at intervals of one-half degree.

24

Ptolemy used the Babylonian sexagesimal numerals and numeral systems (base 60), he did his computations with a standard circle of radius r = 60 units.


The next major contribution to trigonometry came from India.

The first table of sines is found in the ryabhaya.

25


26

Its author, ryabhaa I (c. 475550), used the word ardha-jya for half-chord, which he sometimes turned around to jya-ardha (chord-half); in due time he shortened it to jya or jiva.

Later, when Muslim scholars translated this work into Arabic, they retained the word jiva without translating its meaning.


Thus jiva could also be pronounced as jiba or jaib, and this last word in Arabic means fold or bay.

27


28


When the Arab translation was later translated into Latin, jaib became sinus, the Latin word for bay.

The word sinus first appeared in the writings of Gherardo of Cremona (c. 111487), who translated many of the Greek texts, including the Almagest, into Latin.

29


Other writers followed, and soon the word sinus, or sine, was used in the mathematical literature throughout Europe.

The abbreviated symbol sin was first used in 1624 by Edmund Gunter, an English minister and instrument maker.

The first table of tangents and cotangents was constructed around 860 by abash al-sib (the Calculator), who wrote on astronomy and astronomical instruments.

30


Another Arab astronomer, al-Bttni (c. 858929), gave a rule for finding the elevation of the Sun above the horizon in terms of the length s of the shadow cast by a vertical gnomon of height h.

Al-Bttni's rule, s = h sin (90 )/sin , is equivalent to the formula s = h cot .

31


32

Based on this rule he constructed a table of shadowsessentially a table of cotangentsfor each degree from 1 to 90.

It was through al-Bttni's work that the Hindu half-chord functionequivalent to the modern sinebecame known in Europe.


33

The first definition of a spherical triangle is contained in Book 1 of the Sphaerica, a three-book treatise by Menelaus of Alexandria (c. AD 100) in which Menelaus developed the spherical equivalents of Euclid's propositions for planar triangles.


34

Several Arab scholars, notably Nar al-Dn al-s (120174) and al-Bttni, continued to develop spherical trigonometry and brought it to its present form.

s was the first (c. 1250) to write a work on trigonometry independently of astronomy.


35


But the first modern book devoted entirely to trigonometry appeared in the Bavarian city of Nrnberg in 1533 under the title On Triangles of Every Kind.

Its author was the astronomer Regiomontanus (143676).

36


On Triangles was greatly admired by future generations of scientists; the astronomer Nicolaus Copernicus (14731543) studied it thoroughly, and his annotated copy survives.

37


The final major development in classical trigonometry was the invention of logarithms by the Scottish mathematician John Napier in 1614.

His tables of logarithms greatly facilitated the art of numerical computationincluding the compilation of trigonometry tablesand were hailed as one of the greatest contributions to science.

38


Leonhard Euler

Established the modern trigonometry

Direction of Angles

39

Angles

The opening between two

straight lines drawn from

a single point

The lines are called Sides

The point where they

meet is called Vertex

40

TERMINOLOGY

Adjacent Angles

Two angles having same

Vertex and one common Side

Notation

41

TERMINOLOGY

Coterminal Angles

Two angles have the same

initial and terminal sides

Coterminal angles = 2 -

42

TERMINOLOGY

When to straight lines

meet with other straight

lines as to make two

adjacent equal angles, the

lines are said to be

Perpendiular and each of

the adjacent angles is

called Right Angle

43

TERMINOLOGY

Types of Angles

Acute Angle

Smaller than right angle

Obtuse angle

Greater than right angle but less than two right

angle

44

Types of Angles

Complementary angle

The sum of two angles is equal to right angle

Supplementary angle

The sum of two angles is equal to two right

angles

45

Terminology

46

O

B

r

r A

Chord,

AOB is the angle

subtended at O

by Central Angle,

- Subtended by a

chord

Angle Measures and Unit

Angle is dependent on the direction of the

sides

Unit of Measurement

Degree System

Radian System

Gradient System

47

Degree System First Developed by the Babylonians

Sexagesimal System

Believe that

The four season of the earth repeated

themselves

The sun completed a circuit around the

heavens among the stars in 360 days

1 circle = 360 days = 360 steps or grade

1 circle = 4 season = 4 quadrant

Fourth Quadrant

First Quadrant Second Quadrant

Third Quadrant

48

Degree System

A

1 circle = 6 sextant

1 sextant = 60 degree

1 degree = 60 minute

1 minute = 60 seconds

= =

O

B C

r

r r

49

Radian System

Circular system

O A

B

= =

= ,

,

r

r

=

2

2 =C = 360

=180

1 =180/

50

Gradient System

Conceptualized by the French

1 circle = 400 part =400 Grades

51

Measure of Usual Angles

Right Angle

90 degrees

Straight Angle

180 degrees

First Quadrant Second Quadrant

Fourth Quadrant Third Quadrant

2700

1800

900

00

52

Problem Solving! Area of triangle = base*height

2 = 2 + 2

O

A B

r r

C C

D

2= 2 +2

2 - 2 = 2

= 2 22

----height

Area of triangle = 2c* 2 22

Area of triangle = c* 2 22

53

O

A B

r r

Arc, S

,

, = , 2

2 = 2

S=

Problem Solving!

54

Remember!!

1 circle = 360O = 2 = 400 Grades

55

Sample Problem 1

75 degrees

=________radians

=________grades

=coterminal angles:____________

=supplementary angle:

=complementary angle:

56

Sample Problem 2

350 grades

=________radians

=________degrees


57

Sample Problem 3

/3 grades

=________Grades

=________degrees


=supplementary angle:

=complementary angle:

58

TRIANGLES

Formed by three

intersecting lines at

three points

Three sides

Three angles

59

Part of the triangle

Base

The side where the triangle

supposed to stand

Altitude

A line drawn perpendicular to the

base and through the opposite

vertex.

Base

60

Part of the triangle with respect to

reference angle

Adjacent side

Side near the reference angle

Opposite side

Side opposite to the reference

angle

Hypotenuse (right triangle only)

The longest length of the three

sides

Adjacent

O

p

p

o

s

i

t

e

61

Types of Triangles according to

angles

Right

One of the angles is a

right angle

Oblique

Has no right angle

Obtuse

When one of the angle is obtuse

Acute

If all of the angles are acute

62

Isoceles

Has equal two sides

Equilateral

(equiangular)

Three sides are equal

Scalene

No two sides are

equal

Types of Triangles according to

sides

63

Important Proof

GH is transversal

CHE and BGF or DHE and

AGF are alternate interior

angles

64

Properties of Triangle

65

Pythagorean Theorem

Sum of area of square

66

Trigonometric function of angles

67

Sine

Complementary

Sine

Cosine

Secant Cosecant

Tangent

+ = 90

Remember: Cotangent

Secant: Latin "secant-, secans" from Latin

present participle of "secare" (to cut)

Sine: (jaib) Half-Chord

Tangent: Latin "tangent-, tangens" from

present participle of "tangere" (to touch)


68

sine

cosine

secan

tangent

cosecant

cotangent

sine

cosine

secan

tangent

cosecant

cotangent

Sine

Cosine

Secant Cosecant

Tangent

Cotangent


69

sin =

cos =

tan =

Cosine

Secant

Tangent

Cotangent

Sine

csc =

cot =

sec =

Cosecant

Trigonometric function

and relations

70

sin =

cos =

tan =

csc =

cot =

sec =

sin = 1

csc

= 1

sec

= 1

cot

Special Angle 45O

45O

From Pythagorean Theorem:

c2 = a2 + b2

45O

1

1

b =

= a

c2 = 12 + 12

c = 1 + 1

c = 2

Sin 45O = 1

2

Cos 45O =

1

2

Tan 45O = 1

1 = 1

71

Special Angle 60O, 30O

60O

From Pythagorean Theorem:

= c

= b

12 = a2 + ()2

a2 = 1-1/4 a =3

4

Sin 30O =

1

2

1 = 1/2

Cos 30O =

3

2

1 =

3

2

Tan 30O = 1/2

3

2

= 1

3

60O

30O 30O

1

1 1

1/2 1/2

a =3

2

Sin 60O =

3

2

1 =

3

2

Cos 60O =

1

2

1 = 1

2

Tan 60O =

3

21

2

= 3

c2 = a2 + b2

a

72

Hand Technique

73

Sign Convention

A -

+

-

+

Sine = +

Cosine = + Tangent = +

Sine = +

Cosine = - Tangent = -

Sine = -

Cosine = - Tangent = +

Sine = -

Cosine = + Tangent = -

90O

180O 1,0

0O

270O

-1,0

0,1

0,-1

Unit Circle = radius is 1 A S

T C

74

COFUNCTION THEOREM

75

COFUNCTION IDENTITIES

76

Sample

77

sin30O = cos(90 - 30O)

sin30O = cos(60)

tan x = cot(90 - x) csc 40 = sec (90 - 40)

csc 40 = sec (50)

Even and Odd Function

78

Reference Angle,

Is the acute angle

formed by the

terminal side of and

the horizontal axis

79

Sample

Find the exact value of cos 210O

- Solution: 210 is located at III quadrant

Reference Angle 210 - 180 = 30

cos 30 = 3

2

cos 210 = - 3

2 (210 is located at III quadrant)

80

Sample

Find the exact value of tan 495O

- Solution:

Reference Angle: 495 - 360 = 135

180 - 135 = 45

tan 45 = 1

tan 45 = - 1 (135 is located at II quadrant)

81

Samples

Find the exact values for

82

Circular Functions

83

More Sample

Determine the exact values

84

More Sample

Determine whether the following statements

are true

85

The Graphs of Sinusoidal Functions

The following are examples of things that

repeat in a predictable way

heartbeat

tide levels

time of sunrise

average outdoor temperature for the time of

year

86

Periodic Function

A function f is called a periodic function if there

is a positive number p such that f(x + p) = f(x)

for all x in the domain of f

If p is the smallest such number for which this equation

holds, then p is called the fundamental period

87

sine, cosine, secant, and cosecant functions

have fundamental period of 2 , but that

tangent and cotangent functions have

fundamental period .

Sine Graph

88

Sine Function

89

Cosine Graph

90

Cosine Function

91

PERIOD OF SINUSOIDAL FUNCTIONS

92

Finding the Period of a Sinusoidal Function

y = cos(4x)

y = sin (1/3x)

93

STRATEGY FOR SKETCHING GRAPHS

OF SINUSOIDAL FUNCTIONS

94

Sample

Graph y=3sin(2x)

Step1 A=3, B = 2, p =2/B -----p=

Step2 p/4------- p = /4

Step3

Make a table starting at x=0 to the period x=

in steps of /4

95

Sample

Step3

96

Sample

Step4 Step5

p5

97

Sample

Step6

98

Sample

Graph -2cos(1/3x)

99

Good Luck for the FIRST EXAM

100

Exercise

101

Evaluate the following

expression exactly

Graph the given function

over the given period

Documents

Trigonometry 101