Heights and Distances ? What you’re going to do ? next? 45o
In this situation , the distance or the heights canbe founded by using mathematical techniques,which comes under a branch of ‘trigonometry’.The word ‘ trigonometry’ is derived from theGreek word ‘tri’ meaning three , ‘gon’ meaningsides and ‘metron’ meaning measures.Trigonometry is concerned with the relationshipbetween the angles and sides of triangles. Anunderstanding of these relationships enablesunknown angles and sides to be calculatedwithout recourse to direct measurement.Applications include finding heights/distances ofobjects.
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 Thales of Miletus knowledge of the relationship between the heights of objects 640 – 546 B.C. The first Greek and the length of their shadows to calculate the height for Mathematician. He them. (This will later become the Tangent ratio.) Can you see what predicted the Solar Eclipse of 585 BC. this relationship is, based on the drawings below? h 480 ft 720 ft Similar Similar 6 ft Triangles Triangles 9 ft Sun’s rays casting shadows mid-afternoon Sun’s rays casting shadows late afternoonThales may not have used similar triangles directly to solve the problem but hhe knew that6the ratio of the 6x 720 horizontal sides of each triangle was vertical to h 480 ft sun. Can you use theconstant and unchanging for different heights of the (Egyptian feet of course) 720 9 9measurements shown above to find the height of Cheops?
Later, during the Golden Age of Athens (5 BC.), the philosophers andmathematicians were not particularly interested in the practical side ofmathematics so trigonometry was not further developed. It was another 250 yearsor so, when the centre of learning had switched to Alexandria (current day Egypt)that the ideas behind trigonometry were more fully explored. The astronomer andmathematician, Hipparchus was the first person to construct tables oftrigonometric ratios. Amongst his many notable achievements was hisdetermination of the distance to the moon with an error of only 5%. He used thediameter of the Earth (previously calculated by Eratosthenes) together withangular measurements that had been taken during the total solar eclipse of March190 BC. Eratosthenes Hipparchus of Rhodes 275 – 194 BC 190-120 BC
Early Applications of Trigonometry Finding the height of a mountain/hill. h 25o 20o x dConstructing sundials to Finding the distance toestimate the time from the moon.the sun’s shadow.
Historically trigonometry was developed for work inAstronomy and Geography. Today it is usedextensively in mathematics and many other areas ofthe sciences.•Surveying•Navigation•Physics•Engineering
In this figure, the line ACdrawn from the eye of the Cstudent to the top of thetower is called the line ofsight. The person is lookingat the top of the tower. Theangle BAC, so formed by Towerline of sight with horizontalis called angle of elevation. Angle of elevation A 45o Horizontal level B
A Horizontal level 45o Angle of depressionMountain Object C B
Method of finding the heights or the distances C Tower Angle of elevation A 45o Horizontal level BLet us refer to figure of tower again. If you want tofind the height of the tower i.e. BC without actuallymeasuring it, what information do you need ?
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 thenhow can we determine the height of the height of thetower ? In ∆ABC, the side BC is the opposite side inrelation to the known angle A. Now, which of thetrigonometric ratios can we use ? Which one of themhas the two values that we have and the one we need todetermine ? Our search narrows down to using eithertan A or cot A, as these ratios involve AB and BC. Therefore, tan A = BC/AB or cot A = AB/BC, whichon solving would give us BC i.e., the height of the tower.
Some Applications of trigonometry based on finding heights and distance
Here we have to find the height of the school.Here BC = 28.5 m and AC i.e., the height of theschool = tan 45 = AC/BCi.e., 1 = AC/28.5Therefore , AC = 28.5mSo the height of the school is 28.5 m. A B 45o C 28.5m
Here we have to find the length of the ladder in the below figure and also how far is the foot of the ladder from the house ? (here take √3 = 1.73m)Now, can you think trigonometricratios should we consider ?It should be sin 60So, BC/AB = sin 60 or 3.7/AB =√3/2 BTherefore BC = 3.7 x 2/√3Hence length of the ladder is4.28mNow BC/AC = cot 60 = 1/√3 3.7mi.e., AC = 3.7/√3 = 2.14m (approx) 60oTherefore the foot of theladder from the house is 2.14m. A C
Here we need to find the height of the lighthouse above themountain . Given that AB = 10 m. (here take √3 =1.732). D 10 m B 30o 45o A P
Since we know the height of the mountainis AB so we consider the right ∆PAB. Wehave tan 30 = AB/AP i.e., 1/√3 = 10/APtherefore AP = 10√3m so the distance ofthe building = 10√3m = 17.32mLet us suppose DB = (10+x)m now inright ∆PAD tan 45 = AD/AP = 10+x/10√3therefore 1 = 10+x/10√3 i.e., x = 10(√3-1)=7.32. So, the length of the flagstaff is7.32m
Summary The line of sight is the line drawn from the eye of the observer to the point in the object viewed by the observer. The angle of elevation of an object viewed, 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. The angle of depression of an object viewed, 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 the head to look at the object. The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.