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### Math project some applications of trigonometry

• 2.  Trigonometry is the branch of mathematics that deals with triangles particularly right triangles. For one thing trigonometry works with all angles and not just triangles. They are behind how sound and light move and are also involved in our perceptions of beauty and other facets on how our mind works. So trigonometry turns out to be the fundamental to pretty much everything BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 3.  Angle of Elevation: In the picture below, an observer is standing at the top of a building is looking straight ahead (horizontal line). The observer must raise his eyes to see the airplane (slanting line). This is known as the angle of elevation. BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 4.  Angle of Depression: The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation (this assumes the object is close enough to the observer so that the horizontals for the observer and the object are effectively parallel). BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 5. 2a 2a 600 300 600 a If θ is an angle The 90-θ is it’s complimentary angle 1 1 1  1 1 Sine 45 2 45 2 45    Cos Tan BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 6. A trigonometric function is a ratio of certain parts of a triangle. The names of these ratios are: The sine, cosine, tangent, cosecant, secant, cotangent. Let us look at this triangle… a c b ө A B C Given the assigned letters to the sides and angles, we can determine the following trigonometric functions. The Cosecant is the inversion of the sine, the secant is the inversion of the cosine, the cotangent is the inversion of the tangent. With this, we can find the sine of the value of angle A by dividing side a by side c. In order to find the angle itself, we must take the sine of the angle and invert it (in other words, find the cosecant of the sine of the angle). Sinθ= Cos θ= Tan θ= Side Opposite Hypothenuse Side Adjacent Hypothenuse Side Opposite Side Adjacent = = a c b c = a b BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 7. 60 45 12 h d h 45   1;.. d  h - - - - - - - (1) Substituting (1)..in (2) 1.732; h 12   12  1.732   0.732 16.39 is also equals to d 12 0.732 3 1.732 (2) 12 60            h h h h h d h Tan or d Tan BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 8. 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 30 60 d Tan Tan 1       From d h (1) 3 From h d d (2)  d h h 3( 150)   Substituting the value of d .. .. .. .. .. h h 3( 3 150)   h h 3 150 3   h h 3 150 3 h   2  150 3 h 75*1.732 129.9 m (2) 150 60 3 (1) 3 30          BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 9. Heights and Distances ? 45o ? What you’re going to do next? BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 10. In this situation , the distance or the heights can be founded by using mathematical techniques, which comes under a branch of ‘trigonometry’. The word ‘ trigonometry’ is derived from the Greek word ‘tri’ meaning three , ‘gon’ meaning sides and ‘metron’ meaning measures. Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects. BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 13. Trigonometry An early application of trigonometry was made by Thales on a visit to Egypt. He was surprised that no one could tell him the height of the 2000 year old Cheops pyramid. He used his knowledge of the relationship between the heights of objects and the length of their shadows to calculate the height for them. (This will later become the Tangent ratio.) Can you see what this relationship is, based on the drawings below? Thales of Miletus 640 – 546 B.C. The first Greek Mathematician. He predicted the Solar Eclipse of 585 BC. Similar Triangles h Similar Triangles 6 ft 720 ft 9 ft Sun’s rays casting shadows mid-afternoon Sun’s rays casting shadows late afternoon Thales may not have used similar triangles directly to solve the problem but he knew h that 6  the ratio of the vertical to horizontal sides of each triangle was constant 720 and 9 unchanging for different heights of the sun. Can you use the measurements shown above to find the height of Cheops? 480 ft (Egyptian 4 feet of course) 6 720 9 80 ft x h   BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 14. Later, during the Golden Age of Athens (5 BC.), the philosophers and mathematicians were not particularly interested in the practical side of mathematics so trigonometry was not further developed. It was another 250 years or so, when the centre of learning had switched to Alexandria (current day Egypt) that the ideas behind trigonometry were more fully explored. The astronomer and mathematician, Hipparchus was the first person to construct tables of trigonometric ratios. Amongst his many notable achievements was his determination of the distance to the moon with an error of only 5%. He used the diameter of the Earth (previously calculated by Eratosthenes) together with angular measurements that had been taken during the total solar eclipse of March 190 BC. Hipparchus of Rhodes 190-120 BC Eratosthenes 275 – 194 BC BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 15. Early Applications of Trigonometry h Finding the height of a mountain/hill. Finding the distance to the moon. Constructing sundials to estimate the time from the sun’s shadow. BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 16. Historically trigonometry was developed for work in Astronomy and Geography. Today it is used extensively in mathematics and many other areas of the sciences. •Surveying •Navigation •Physics •Engineering BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 17. 45o Angle of elevation A C B In this figure, the line AC drawn from the eye of the student to the top of the tower is called the line of sight. The person is looking at the top of the tower. The angle BAC, so formed by line of sight with horizontal is called angle of elevation. Tower Horizontal level BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 18. 45o Mountain Angle of depression A B Object C Horizontal level BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 19. Method of finding the heights or the distances 45o Angle of elevation A C B Tower Horizontal level Let us refer to figure of tower again. If you want to find the height of the tower i.e. BC without actually measuring it, what information do you need ? BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B
• 20. We would need to know the following: i. The distance AB which is the distance between tower and the person . ii. The angle of elevation angle BAC . Assuming that the above two conditions are given then how can we determine the height of the height of the tower ? In ΔABC, the side BC is the opposite side in relation to the known angle A. Now, which of the trigonometric ratios can we use ? Which one of them has the two values that we have and the one we need to determine ? Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC. Therefore, tan A = BC/AB or cot A = AB/BC, which on solving would give us BC i.e., the height of the tower. BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B