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MATHS PRESENTATION

SOME APPLICATIONSOFTRIGONOMETRY

BY:Cdt.T.Navilan SidharthX std. B sec.ROLL NO : 6087PALLAVA HOUSESAINIK SCHOOL AMARAVATHINAGAR

USES OF TRIGONOMETRY

Trigonometry is mainly used in astronomy to measure distances of various stars. It is also used in measurement of heights of mountains, buildings, monument, etc..

The knowledge of trigonometry also helps us to construct maps,determine the position of an island in relation to latitudes, longitudes

THEODOLITE

It is a surveying instrument on the basis of Trigonometry .

THEODOLITE AND Mt EVERESTDuring a survey in 1856 of British India the highest peak from a distance of over 60 km this 8848 m high peak located in Nepal is named Everest after Sir George Everest. Who used the first Theodolite . This instrument is now in MUSEUM OF SURVEYS

Line of Sight

HorizontalLine of Sight

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We observe generally that children usually look up to see an aero plane when it passes overhead. This line joining their eye to the plane , while looking up is called Line of sight

Angle of Elevation

The angle which the line of sight makes with a horizontal line drawn away from their eyes is called the angle of Elevation of aero plane from them.Angle of Elevation

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 parallelAngle of Depression

Early Beginning uses of trigonometry for determining heights and distances

Trigonometry (Three-angle-measure)

The Great Pyramid (Cheops) at Giza, near Cairo, one of the 7 wonders of the ancient word. (The only one still surviving).This is the one of the earliest use of trigonometry. People use trigonometry for determining height of this pyramid.

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.

Eratosthenes275 194 BCHipparchus of Rhodes 190-120 BCThe library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained 600 000 manuscripts.

Early Applications of Trigonometry

Finding the height of a mountain/hill.

Constructing sundials to estimate the time from the suns shadow.

Finding the distance to the moon.

h

Historically trigonometry was developed for work in Astronomy and Geography. Today it is used extensively in mathematics and many other areas of the sciences.SurveyingNavigationPhysicsEngineering

Now let us Solve some problem related toHeight and Distance

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.

Let AB be the tower and the angle of elevation from point C (on ground) is 30.In ABC,

Therefore, the height of the tower is = 17.32

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 ABC,

Therefore, the height of the pole is 10 m.

A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60. Find the length of the string, assuming that there is no slack in the string.

Let K be the kite and the string is tied to point P on the ground.In KLP,

Hence, the length of the string is =70.18

THANK YOU

BY:T.NAVILAN SIDHARTH