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Trigonometry101
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- 2. • Consultation Time:2:30 – 4:30 PM • Main Book: – Algebra and Trigonometry by Loius Liethold • Reference Book:
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- 4. UM Core Values Excellence Honestyand Integrity Teamwork Innovation
- 5. Course Description Trigonometric functions;identities and equations; solutions of triangles; law of sines; law of cosines; inverse trigonometric functions; spherical trigonometry 5
- 6. Course Objectives After completingthis 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
- 7. Course Outline 1. TrigonometricFunctions 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
- 8. Course Outline 2. AnalyticTrigonometry 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
- 9. Course Outline 3. Applicationof 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
- 10. TRIGONOMETRY • A branchof 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
- 11. 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
- 12. 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
- 13. Application of Trigonometry •Carpentry • Mechanics • Machine work • Astronomy • Land survey and measurement • Map making, • Artillery range finding. • And others 13
- 14. • Greek Word(origin) 14 History of TRIGONOMETRY
- 15. 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. 16 • The Rhindpapyrus, 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. 17 For example, problem56 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
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- 19. 19 • Trigonometry beganwith 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. 20 • He consideredevery 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. 21 • To computethe 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. 22 • The firstmajor 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. 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. 24 • Ptolemy usedthe Babylonian sexagesimal numerals and numeral systems (base 60), he did his computations with a standard circle of radius r = 60 units. History of Trigonometry
- 25. • The nextmajor contribution to trigonometry came from India. • The first table of sines is found in the Āryabhaṭīya. 25 History of Trigonometry
- 26. 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
- 27. • Thus jivacould also be pronounced as jiba or jaib, and this last word in Arabic means “fold” or “bay.” 27 History of Trigonometry
- 28. 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. 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.
- 30. • The firsttable 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
- 31. • Another Arabastronomer, 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. 32 • Based onthis 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. 33 • The firstdefinition 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. 34 • Several Arabscholars, 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. 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. 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. 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. 38 History of Trigonometry •Leonhard Euler – Established the modern trigonometry
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- 40. • Angles – Theopening between two straight lines drawn from a single point • The lines are called Sides • The point where they meet is called Vertex 40 TERMINOLOGY
- 41. • Adjacent Angles –Two angles having same Vertex and one common Side • Notation 41 TERMINOLOGY
- 42. • Coterminal Angles –Two angles have the same initial and terminal sides – Coterminal angles = 2π - θ 42 TERMINOLOGY
- 43. • When tostraight 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
- 44. Types of Angles •Acute Angle – Smaller than right angle • Obtuse angle – Greater than right angle but less than two right angle 44
- 45. 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
- 46. Terminology 46 O B r r A Chord, 𝐴𝐵 𝜃 • ∠AOBis the angle subtended at O by 𝐴𝐵 • Central Angle, 𝜃 - Subtended by a chord
- 47. Angle Measures andUnit • Angle is dependent on the direction of the sides • Unit of Measurement – Degree System – Radian System – Gradient System 47
- 48. Degree System • FirstDeveloped 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 QuadrantSecond Quadrant Third Quadrant 48
- 49. Degree System A 1 circle= 6 sextant 1 sextant = 60 degree 1 degree = 60 minute 1 minute = 60 seconds 𝑂𝐵 =𝑂𝐴 =𝐴𝐵 O BC r r r 49
- 50. Radian System • Circularsystem 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
- 51. Gradient System • Conceptualizedby the French – 1 circle = 400 part =400 Grades 51
- 52. Measure of UsualAngles • Right Angle – 90 degrees • Straight Angle – 180 degrees First QuadrantSecond Quadrant Fourth QuadrantThird Quadrant 2700 1800 900 00 52
- 53. Problem Solving! Area oftriangle = ½ base*height 𝑂𝐴2 = 𝑂𝐷2 + 𝐴𝐷2 O AB rr 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
- 54. O AB rr 𝜃 Arc, S 𝐴𝑟𝑐 ,𝑆 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒,𝐶 = 𝐴𝑛𝑔𝑙𝑒,𝜃 2𝜋 𝑆 2𝜋𝑟 = 𝜃 2𝜋 S=𝑟𝜃 Problem Solving! 54
- 55. Remember!! 1 circle =360O = 2𝜋 = 400 Grades 55
- 56. Sample Problem 1 •75 degrees =________radians =________grades =coterminal angles:____________ =supplementary angle: =complementary angle: 56
- 57. Sample Problem 2 •350 grades =________radians =________degrees =coterminal angles:____________ 57
- 58. Sample Problem 3 •π/3 grades =________Grades =________degrees =coterminal angles:____________ =supplementary angle: =complementary angle: 58
- 59. TRIANGLES • Formed bythree intersecting lines at three points • Three sides • Three angles 59
- 60. Part of thetriangle • Base – The side where the triangle supposed to stand • Altitude – A line drawn perpendicular to the base and through the opposite vertex. Base 60
- 61. Part of thetriangle 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
- 62. Types of Trianglesaccording 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
- 63. • Isoceles – Hasequal two sides • Equilateral (equiangular) – Three sides are equal • Scalene – No two sides are equal Types of Triangles according to sides 63
- 64. Important Proof • GHis transversal • CHE and BGF or DHE and AGF are alternate interior angles 64
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- 66. Pythagorean Theorem • Sumof area of square 66
- 67. Trigonometric function ofangles 67 Sine𝛼 Complementary Sine “Cosine” Secant Cosecant Tangent𝛽 𝛽 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒 𝛼 + 𝛽 = 90 𝑂 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. Trigonometric function ofangles 68 sine 𝛼 cosine 𝛼 secan𝑡 𝛼 tangent 𝛼 cosecant 𝛼 cotangent 𝛼 sine 𝛽 cosine 𝛽 secan𝑡 𝛽 tangent 𝛽 cosecant 𝛽 cotangent 𝛽 𝛼 Sine “Cosine” Secant Cosecant Tangent𝛽 𝛽 Cotangent
- 69. Trigonometric function ofangles 69 sin 𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠 cos 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠 tan 𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝛼 “Cosine” Secant Tangent𝛽 𝛽 Cotangent Sine csc 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 cot 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 sec 𝛼 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 Cosecant
- 70. Trigonometric function and relations 70 sin𝛼 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠 cos 𝛼 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑎𝑛𝑒𝑜𝑢𝑠 tan 𝛼 = 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
- 71. Special Angle 45O 45O FromPythagorean 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
- 72. 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 11 1/21/2 a = 3 2 Sin 60O = 3 2 1 = 3 2 Cos 60O = 1 2 1 = 1 2 Tan 60O = 3 2 1 2 = 3 c2 = a2 + b2 a 72
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- 74. 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 1AS T C 74
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- 77. Sample 77 sin30O = cos(90°- 30O) sin30O = cos(60°) tan x = cot(90° - x) csc 40 = sec (90° - 40) csc 40 = sec (50°)
- 78. Even and OddFunction 78
- 79. Reference Angle, θ’ •Is the acute angle formed by the terminal side of and the horizontal axis 79
- 80. Sample • Find theexact 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
- 81. Sample • Find theexact 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
- 82. Samples • Find theexact values for 82
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- 84. More Sample • Determinethe exact values 84
- 85. More Sample • Determinewhether the following statements are true 85
- 86. The Graphs ofSinusoidal 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
- 87. Periodic Function • Afunction 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 .
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- 92. PERIOD OF SINUSOIDALFUNCTIONS 92
- 93. Finding the Periodof a Sinusoidal Function • y = cos(4x) • y = sin (1/3x) 93
- 94. STRATEGY FOR SKETCHINGGRAPHS OF SINUSOIDAL FUNCTIONS 94
- 95. Sample Graph y=3sin(2x) • Step1A=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
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- 100. • Good Luckfor the FIRST EXAM 100
- 101. Exercise 101 Evaluate the following expressionexactly Graph the given function over the given period