# Heights & distances

23 de Jan de 2016
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### Heights & distances

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• 3. 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. 3
• 4. Some historians say that trigonometry was invented by Hipparchus, a Greek mathematician. He also introduced the division of a circle into 360 degrees into Greece. 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. 4
• 5. History of trigonometry  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).
• 6. B A C Sin / Cosec  P (pandit) H (har) Cos / Sec  B (badri) H (har) Tan / Cot  P (prasad) B (bole) BASE (B) PERPENDICULAR (P)  6
• 7. 7 00 300 450 600 900 Sine 0 1/2 1/2 3/2 1 Cosine 1 3/2 1/2 1/2 0 Tangent 0 1/ 3 1 3 Not defined Cosecant Not defined 2 2 2/ 3 1 Secant 1 2/ 3 2 2 Not defined Cotangent Not defined 3 1 1/ 3 0
• 8. 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  8
• 9. Applications 9 Measuring inaccessible lengths Height of a building (tree, tower, etc.) Width of a river (canyon, etc.)
• 10. 10  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
• 11. 11  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
• 12. 12 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
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• 14. 1. The angle of elevation of the top of a tower from a point At the foot of the tower is 300 . And after advancing 150mtrs Towards the foot of the tower, the angle of elevation becomes 600 .Find the height of the tower? 150 h d 30 60 mh h hh hh hh dofvaluethengSubstituti hd From hdFrom d h Tan d h Tan 9.129732.1*75 31502 31503 31503 )1503(3 .......... )150(3 )2( 3)1( )2( 150 360 )1( 3 1 30           
• 15. 15 2.The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 30°. Find the height of the tower. Therefore, the height of the tower is
• 16. 16 3. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30 °.Sol:- It can be observed from the figure that AB is the pole. In ΔABC, Therefore, the height of the pole is 10 m.
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• 19. 19 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!
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