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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
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 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
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.
History of Trigonometry
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.
History of Trigonometry
18
History of Trigonometry
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
History of Trigonometry
20
• 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.
History of Trigonometry
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.
History of Trigonometry
22
• 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.
History of Trigonometry
23
History of Trigonometry
• 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.
History of Trigonometry
• The next major contribution to trigonometry came from India.
• The first table of sines is found in the Āryabhaṭīya.
25
History of Trigonometry
26
• 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.
History of Trigonometry
• Thus jiva could also be pronounced as jiba or jaib, and this last word in Arabic means “fold” or “bay.”
27
History of Trigonometry
28
History of Trigonometry
• 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.
29
History of Trigonometry
• 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.
30
History of Trigonometry
• 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.
31
History of Trigonometry
32
• 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.
History of Trigonometry
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.
History of Trigonometry
34
• 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.
History of Trigonometry
35
History of Trigonometry
• 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).
36
History of Trigonometry
• On Triangles was greatly admired by future generations of scientists; the astronomer Nicolaus Copernicus (1473–1543) studied it thoroughly, and his annotated copy survives.
37
History of Trigonometry
• 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.
38
History of Trigonometry
• 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 = 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
45
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
48
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
49
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
55
Sample Problem 1
• 75 degrees
=________radians
=________grades
=coterminal angles:____________
=supplementary angle:
=complementary angle:
56
Sample Problem 2
• 350 grades
=________radians
=________degrees
=coterminal angles:____________
57
Sample Problem 3
• π/3 grades
=________Grades
=________degrees
=coterminal angles:____________
=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
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
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 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)
Trigonometric function of angles
68
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
Trigonometric function of angles
69
sin 𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠
cos 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠
tan𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝛼
“Cosine”
Secant
Tangent 𝛽
𝛽
Cotangent
Sine
csc 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
cot 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
sec 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Cosecant
Trigonometric function
and relations
70
sin 𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠
cos𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠
tan𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
csc 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
cot 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
sec 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
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
71
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)
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 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
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