1. 1. Power Point Presentation on Introduction of Trigonometry By:- Kumar Prashant Dwivedi M.sc(Physics), B.Ed.
2. 2. Level:- 10th Subject:- maths Chapter:- Introduction of Trigonometry No. of period required:- 10
3. 3. Lets take some examples from our surroundings where right triangle can be imagined to form:- If a student is looking at the top of the tower , a right triangle can be imagined to be made
4. 4. Half Slice of bread can be imagined to be made a right triangle
5. 5. In all the situation distances or heights can be found by using some mathematical techniques which comes under a branch of mathematics called trigonometry. Trigonometry is derived from Greek words Tri + gon + metron (three) (sides) (measure) In fact Trigonometry is the study of relation between the sides and angles of a triangle.
6. 6. Historical Background The history of trigonometry dates back to the early age of Egypt and Babylon. Angles were then measured in degree. It was then advanced by the Greek astronomer Hipperchus in the second century of B.C who compiled a trigonometric table that measured the length of chord subtending a various angles in a circle of a fixed radius r.
7. 7. Hipparchus is considered the greatest astronomical observer, and by some the greatest astronomer of antiquity. He was the first Greek to develop quantitative and accurate models for the motion of the Sun and Moon. With his solar and lunar theories and his numerical trigonometry, he was probably the first to develop a reliable method to predict solar eclipses.
8. 8. Key Concepts Trigonometric ratios Trigonometric ratios at specific angles (0, 30, 45, 60, 90) Trigonometric ratios at complementary angles Trigonometric Identities Angle of Elevation Angle of Depression
9. 9. Methodology Previous knowledge testing Demonstration Method Explanation by giving examples Relating to daily life Concrete to abstract thinking Simple to complex Learning by doing
10. 10. B A C Sin / Cosec P (pandit) H (har) Cos / Sec B (badri) H (har) Tan / Cot P (prasad) B (bole) This is pretty easy! BASE (B) PERPENDICULAR (P)
11. 11. A 0 30 45 6 0 90 Sin A 0 1 Cos A 1 0 Tan A 0 1 Not Defined Cosec A Not Defined 2 1 Sec A 1 2 Not Defined Cot A Not Defined 1 0
12. 12. Trigonometric Ratio of complementary Angles sin( 90 - ) = cos cos( 90- ) = sin tan( 90 - ) = cot cot( 90 - ) = tan sec( 90 - ) = cosec cosec( 90 - ) = sec
13. 13. Sin2 + Cos2 = 1 1 Cos2 = Sin2 1 Sin2 = Cos2 Tan2 + 1 = Sec2 Sec2 - Tan2 = 1 Sec2 - 1 = Tan2 Cot2 + 1 = Cosec2 Cosec2 - Cot2 = 1 Cosec2 - 1 = Cot2 13
14. 14. Vocabulary / Terminology Used Right Triangle Base, perpendicular and hypotenuse Sine ,Cosine, Tangent, Cotangent, Secant and Cosecant Elevation Depression
15. 15. Life Skills Integrated Identify and problem solving Attitudes It enhance Scientific Skill which improves critical thinking.
16. 16. Historically, it was developed for astronomy and geography, but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, trigonometry is used in physics, engineering, and chemistry. Within mathematics, trigonometry is used primarily in calculus (which is perhaps its greatest application), linear algebra, and statistics. Since these fields are used throughout the natural and social sciences, trigonometry is a very useful subject to know.
17. 17. Applications Measuring inaccessible lengths Height of a building (tree, tower, etc.) Width of a river (canyon, etc.) Angle of elevation and depression 17
18. 18. 18 Angle of Elevation It is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e., the case when we raise our head to look at the object. A HORIZONTAL LEVEL ANGLE OF ELEVATION
19. 19. 19 Angle of Depression It is the angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e., the case when we lower our head to look at the object. A HORIZONTAL LEVEL ANGLE OF DEPRESSION
20. 20. Application: Height To establish the height of a building, a person walks 120 ft away from the building. At that point an angle of elevation of 30 is formed when looking at the top of the building. 20 30 120 ft h = ? H = 69.28 ft
21. 21. Application: Height An observer on top of a hill measures an angle of depression of 60 when looking at a truck parked in the valley below. If the truck is 55 ft from the base of the hill, how high is the hill? 21 60 h = ? 55 ft H = 95.26 ft
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23. 23. 23 ? 70 ft 30 D = 40.41 ft
24. 24. 24 h = ? HORIZONTAL LEVEL It is an instrument which is used to measure the height of distant objects using trigonometric concepts. Here, the height of the tree using T. concepts, h = tan *(x) x units
25. 25. Triangle Jokes Trigonometry jokes are a sine of the times. Fake tan: The major threat to trigonometry Q. What does trigonometry have in common with a beach? A: Tan Gents Trigonometry for farmers: swine and coswine. When were trigonometry tables used? B. C., Before Calculators.
26. 26. 26 Trigonometry begins in the right triangle, but it doesnt have to be restricted to triangles. The trigonometric functions carry the ideas of triangle trigonometry into a broader world of real-valued functions and wave forms. Trig functions are the relationships amongst various sides in right triangles. The enormous number of applications of trigonometry include astronomy, geography, optics, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, seismology, land surveying, architecture. I get it!
27. 27. The Pyramids of Giza Primitive forms of trigonometry were used in the construction of these wonders of the world.
28. 28. Architecture In architecture, trigonometry plays a massive role in the compilation of building plans. For example, architects would have to calculate exact angles of intersection for components of their structure to ensure stability and safety. Some instances of trigonometric use in architecture include arches, domes, support beams, and suspension bridges. Architecture remains one of the most important sectors of our society as they plan the design of buildings and ensure that they are able to withstand pressures from inside.
29. 29. How do I get involved in Architecture? Classes to TAke Salary and Benefits Physics Geometry Trigonometry Pre-Calculus and Calculus Engineering 3-D Design Drawing * Art classes will assist you in being able to conceptualize objects! Most architects start out with a salary of Rs 371268 + per year , and through experience, may earn up to Rs 500000 or above Becoming an architect will open you to many more careers, including interior design and building design!
30. 30. Jantar Mantar observatory For millenia, trigonometry has played a major role in calculating distances between stellar objects and their paths.
31. 31. Astronomy Astronomy has been studied for millennia by civilizations in all regions of the world. In our modern age, being able to apply Astronomy helps us to calculate distances between stars and learn more about the universe. Astronomers use the method of parallax, or the movement of the star against the background as we orbit the sun, to discover new information about galaxies. Menelaus Theorem helps astronomers gather information by providing a backdrop in spherical triangle calculation.
32. 32. How do I get involved in Astronomy? Classes to take Salary and Benefits Physics Electronics Advanced Math Geometry Precalculus and Calculus Astrophysics The median salary for Astronomers is salary packets ranging from Rs. 8 lacs per annum to around Rs. 10 lacs per annum for those in senior positions. As an aside to being an astronomer, one can also acquire a teaching position at a research university!
33. 33. Grand Canyon Skywalk Geologists had to measure the amount of pressure that surrounding rocks could withstand before constructing the skywalk.
34. 34. Geology Trigonometry is used in geology to estimate the true dip of bedding angles. Calculating the true dip allows geologists to determine the slope stability. Although not often regarded as an integral profession, geologists contribute to the safety of many building foundations. Any adverse bedding conditions can result in slope failure and the entire collapse of a structure.
35. 35. How do I get involved in Geology? Classes to take Salary and Benefits Physics Chemistry Pre-calculus and Calculus Geometry Geochemistry Seismology Median wages for Geologists are around Rs 500000+ a year. However, if involved in oil extraction, earnings could increase to over Rs 700000+ a year. Geologists can be very flexible in what they decide to do. There are a multitude of job options ranging from agriculture to tourism that require the work of a geologist.
36. 36. Reference offline/online:- NCERT text book of mathematics www.britanica.com www.mathsfun.com http://math.lotsoflessons.com
37. 37. Home Assignment 1. Related Exercises from text book and exemplar book of mathematics. 2. Activity:- to prepare a chart of T-ratios, T-ratios at specific angles, T-identities. 3. Project:- make clinometer ( in group)
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50. 50. Remedial Plan Make awareness regarding the ratio by giving examples from daily life. Group leaders will be assigned to make correction by guiding them. Similar problem assigned to practices related to each concept will be given.
51. 51. Thanks & Have A Nice Day