Trigonometry101

TRIGONOMETRY MATH 102

2ndSEM/SY2014-2015

• Consultation Time: 2:30 – 4:30 PM

• Main Book:

– Algebra and Trigonometry by Loius Liethold

• Reference Book:

VMG

UM Core Values

Excellence

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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

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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 civilizations—in particular, the Egyptian, Babylonian, Hindu, and Chinese—possessed a considerable knowledge of practical geometry, including some concepts that were a prelude to trigonometry.

15

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• 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.


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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.


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19

• Trigonometry began with the

Greeks.

• Hipparchus (c. 190–120 BC) was

the first to construct a table of

values for a trigonometric function. – Astronomer

– founder of trigonometry


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• He considered every triangle—planar or spherical—as 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 it—or, equivalently, the length of a chord as a function of the corresponding arc width.


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• The first major ancient work on trigonometry to reach Europe intact after the Dark Ages was the Almagest by Ptolemy (c. AD 100–170).

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


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• 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.

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


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• The next major contribution to trigonometry came from India.

• The first table of sines is found in the Āryabhaṭīya.

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• Its author, Āryabhaṭa I (c. 475–550), 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.”

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• 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. 1114–87), who translated many of the Greek texts, including the Almagest, into Latin.

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• 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 Ḥabash al-Ḥāsib (“the Calculator”), who wrote on astronomy and astronomical instruments.

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• Another Arab astronomer, al-Bāttāni (c. 858–929), gave a rule for finding the elevation θ of the Sun above the horizon in terms of the length s of the shadow cast by a vertical gnomon of height h.

• Al-Bāttāni's rule, s = h sin (90° − θ)/sin θ, is equivalent to the formula s = h cot θ.

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• Based on this rule he constructed a “table of shadows”—essentially a table of cotangents—for each degree from 1° to 90°.

• It was through al-Bāttāni's work that the Hindu half-chord function—equivalent to the modern sine—became known in Europe.


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• 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.





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• Several Arab scholars, notably Naṣīr al-Dīn al-Ṭūsī (1201–74) and al-Bāttāni, continued to develop spherical trigonometry and brought it to its present form.

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


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

• Its author was the astronomer Regiomontanus (1436–76).



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• On Triangles was greatly admired by future generations of scientists; the astronomer Nicolaus Copernicus (1473–1543) studied it thoroughly, and his annotated copy survives.

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• 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 computation—including the compilation of trigonometry tables—and were hailed as one of the greatest contributions to science.

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• Leonhard Euler

– Established the modern trigonometry

Direction of Angles

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• 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 = 2π - θ

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

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Terminology

46

O

B

r

r A

Chord, 𝐴𝐵

𝜃

• ∠AOB is the

angle

subtended at O

by 𝐴𝐵 • Central Angle, 𝜃

- 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

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Degree System

A

1 circle = 6 sextant

1 sextant = 60 degree

1 degree = 60 minute

1 minute = 60 seconds

𝑂𝐵 =𝑂𝐴 =𝐴𝐵

O

B C

r

r r

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Radian System

• Circular system

O A

B

𝑂𝐵 =𝑂𝐴 = 𝐴𝐵

𝜋 = 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒, 𝐶

𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟, 𝐷

r

r

𝜋 = 𝐶

2𝑟

2𝑟𝜋 =C = 360

𝑟𝜋 =180

1 𝑟𝑎𝑑𝑖𝑎𝑛 =180/𝜋

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 = ½ base*height

𝑂𝐴2 = 𝑂𝐷2 + 𝐴𝐷2

O

A B

r r

𝜃

C C

D

𝑟2= 𝑂𝐷2 +𝑐2

𝑟2 - 𝑐2 = 𝑂𝐷2

𝑂𝐷 = 𝑟2 − 𝑐22

----height

Area of triangle = ½ 2c* 𝑟2 − 𝑐22

Area of triangle = c* 𝑟2 − 𝑐22

53

O

A B

r r 𝜃

Arc, S

𝐴𝑟𝑐 ,𝑆

𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒,𝐶 = 𝐴𝑛𝑔𝑙𝑒,𝜃 2𝜋

𝑆

2𝜋𝑟 = 𝜃 2𝜋

S= 𝑟𝜃

Problem Solving!

54

Remember!!

1 circle = 360O = 2𝜋 = 400 Grades

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Sample Problem 1

• 75 degrees

=________radians

=________grades

=coterminal angles:____________

=supplementary angle:

=complementary angle:

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Sample Problem 2

• 350 grades

=________radians

=________degrees


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Sample Problem 3

• π/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

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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

𝛼

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

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• 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 𝛼

Complementary

Sine

“Cosine”

Secant Cosecant

Tangent 𝛽

𝛽

𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒 𝛼 + 𝛽 = 90𝑂

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)


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sine 𝛼

cosine𝛼

secan𝑡 𝛼

tangent 𝛼

cosecant 𝛼

cotangent 𝛼

sine 𝛽

cosine𝛽

secan𝑡 𝛽

tangent 𝛽

cosecant 𝛽

cotangent 𝛽

𝛼 Sine

“Cosine”

Secant Cosecant

Tangent 𝛽

𝛽

Cotangent


69

sin 𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠

cos 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡


tan𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝛼

“Cosine”

Secant

Tangent 𝛽

𝛽

Cotangent

Sine

csc 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

cot 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡


sec 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒


Cosecant

Trigonometric function

and relations

70

sin 𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒


cos𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡


tan𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒


csc 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒


cot 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡


sec 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒


sin 𝛼 = 1

csc 𝛼

𝑐𝑜𝑠 𝛼 = 1

sec 𝛼

𝑡𝑎𝑛 𝛼 = 1

cot 𝛼

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

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Special Angle 60O, 30O

60O

From Pythagorean Theorem:

= c

= b

12 = a2 + (½)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, θ’

• 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)

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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 2π , but that

tangent and cotangent functions have

fundamental period π .

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 =2π/B -----p=π

• Step2 p/4------- p = π/4

• Step3

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

in steps of π/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

Engineering

Trigonometry101