Learning network activity 3
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Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles, especially right triangles. It has many applications in fields like astronomy, navigation, engineering, and more. Some key uses of trigonometry include measuring inaccessible heights and distances by using trigonometric functions and properties of triangles formed by angles of elevation or depression. For example, trigonometry can be used to calculate the height of a building or tree by measuring the angle of elevation from a known distance away. It also has applications in measuring distances in astronomy, designing curved architectural structures, and calculating road grades. The document provides examples of various real-world applications of trigonometric concepts.
Learning network activity 3
- 3. Introduction of Trigonometry Trigonometry is derived from Greek words trigonon (three angles) and metron ( measure). Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees Triangles on a sphere are also studied, in spherical trigonometry. Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, on the trigonometric functions, and with calculations based on these functions.
- 4. Trigonometry is a branch of Mathematics that deals with the distances or heights of objects which can be found using some mathematical techniques. The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three) , ‘gon’ (meaning sides) and ‘metron’ (meaning measure). 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.
- 5. History • • • • • The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago. Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly compute the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle). Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al-Kashi made significant contributions in this field(trigonometry).
- 6. C PERPENDICULAR (P) B Sin / Cosec Cos / Sec Tan / Cot P (pandit) B (badri) P (prasad) H (har) A BASE (B) B (bole) H (har) This is pretty easy! 6
- 7. Three Types Trigonometric Ratios There are 3 kinds of trigonometric ratios we will learn. sine ratio cosine ratio tangent ratio
- 8. Right Triangle A triangle in which one angle is equal to 90 is called right triangle. The side opposite to the right angle is known as hypotenuse. AB is the hypotenuse The other two sides are known as legs. AC and BC are the legs Trigonometry deals with Right Triangles
- 9. Pythagoras Theorem In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs. In the figure AB2 = BC2 + AC2
- 10. Trigonometry C Hypotenuse O x0 Opposite The ‘Hypotenuse’ is always opposite the right angle Adjacent F y0 C Hypotenuse O Opposite The ‘Opposite’ is always opposite the angle under investigation. The ‘Adjacent’ is always alongside the angle under investigation. Adjacent F
- 11. sin opposite side hypotenuse cos adjacent side hypotenuse tan opposite side adjacent side Make Sure that the triangle is right-angled
- 12. Relation between different Trigonometric Identities • Sine • Cosine • Tangent • Cosecant • Secant • Cotangent
- 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
- 14. A 0 30 45 60 90 Sin A 0 1 Cos A 1 0 Tan A 0 Cosec A Not Defined Sec A Not Defined Not Defined 2 1 1 Cot A 1 2 1 Not Defined 0 14
- 15. How can trigonometry be used to solve realworld problems?
- 16. Applications of Trigonometry • This field of mathematics can be applied in astronomy,navigation, music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development. 16
- 17. Applications • Measuring inaccessible lengths • • Height of a building (tree, tower, etc.) Width of a river (canyon, etc.)
- 18. 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 32 is formed when looking at the top of the building. h=? 32 120 ft H = 74.98 ft 18
- 19. Application: Height 68 • • An observer on top of a hill measures an angle of depression of 68 when looking at a truck parked in the valley below. h=? If the truck is 55 ft from the base of the hill, how high is the hill? 55 ft H = 136.1 ft 19
- 20. • 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. HORIZONTAL LEVEL A ANGLE OF DEPRESSION 20
- 21. • 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 ANGLE OF ELEVATION HORIZONTAL LEVEL 21
- 22. Angles of Elevation and Depression Line of sight: The line from our eyes to the object, we are viewing. Angle of Elevation:The angle through which our eyes move upwards to see an object above us. Angle of depression:The angle through which our eyes move downwards to see an object below us. 22
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- 24. ? 37 70 ft D = 52.7 ft 24
- 25. • Road has a grade of 5.5%. • Convert this to an angle expressed in degrees. 5.5 ft ? 100 ft A = 3.1 25
- 26. Applications of Trigonometry in Astronomy • • • • Since ancient times trigonometry was used in astronomy. The technique of triangulation is used to measure the distance to nearby stars. In 240 B.C., a mathematician named Eratosthenes discovered the radius of the Earth using trigonometry and geometry. In 2001, a group of European astronomers did an experiment that started in 1997 about the distance of Venus from the Sun. Venus was about 105,000,000 kilometers away from the Sun .
- 27. • • Application of Trigonometry in Architecture Many modern buildings have beautifully curved surfaces. Making these curves out of steel, stone, concrete or glass is extremely difficult, if not impossible. • One way around to address this problem is to piece the surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create what looks like a curved surface. • The more regular these shapes, the easier the building process. • Regular flat shapes like squares, pentagons and hexagons, can be made out of triangles, and so trigonometry plays an important role in architecture. 27
- 28. Trigonometry begins in the right triangle, but it doesn’t 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! 28
- 29. Reference list Niona. N. (2011). Trigonometry. (online), available:http://www.slideshare.net/krunittayamath.06 december 2011. accessed by thabile on 05 march 2014. Pete013 .( 2013). Trigonometry.(online), available: http://www.slideshare.net/pete013?utm_campaign=profiletracking&utm_medium=sssite&utm_source=ssslidevie w . 04 november 2013. Accessed by Thabile on 05 March 2014 Watts. M. (2011). Real world application of trigonometry. (online), available:http://www.slideshare.net/m42watts?utm_campaign=profiletracking&utm_medium=sssite&utm_source =ssslideview. 23 june 2011. Accessed by Thabile on 05 march 2014. 29
- 30. THANK YOU ! Name : thabile 30