Applications of

TrigonometryBy AKSHAT GOYAL

X-C

1030-c

What is Trigonometry?Trigonometry' (from Greek trigōnon ”triangle ”+ metron ”measure”) is a branch of mathematics that studies the relationship of lengths and angles in triangles. The field emerged during the third century BC, evolving out of a understanding of geometry then being used extensively for astronomical studies. The 3rd century astronomers first noted that the lengths of the sides of a right angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics.

History of TrigonometryClassical Greek mathematicians (such as Euclid and Archimedes)studied the Properties of chords and inscribed angles in circles, and proved Theorems that are equivalent to modern trigonometricformulae, although they presented Them geometrically rather than algebraically. Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.

The Egyptians used a primitive form of Trigonometry for buildingpyramids in the 2nd millennium BC.

The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry.

The next significant developments of trigonometry were in India. Influential works from the 4th–5th century, known as the Siddhantas first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine and inverse sine.Trigonometry was still so little known in 16th-century Europe that Nicolaus Copernicus devoted two chapters of “De revolutionibus orbium coelestium” to explain its basic concepts.

ARYABHATTA

NICOLAUS

Applications of TrigonometryThere is an enormous number of applications of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.

In the following slides, we will learn what is line of sight, angle of elevation, angle of depression, and also solve some problems related to trigonometry using trigonometric ratios.

Line of sight , Angle of Elevation and

Angle of Depression

Suppose a boy is looking at a bird on a tree, so the line joining the eye of the boy and the bird is called the Line of Sight.Lets take the same case again that a boy is looking at a bird on a tree. The angle which the line of sight makes with a horizontal line drawn away from the eyes is called the angle of elevation.Now if we consider that the bird is looking at the boy, then the angle between the bird’s line of sight and horizontal line drawn from its eyes is called the Angle of Depression.

Angle of Depression

Line of SightAngle of Elevation

Examples…A man is standing at a distance from a building of height 30 m. The angle of elevation from the man’s eyes to the top of the tower is 45 degrees.Find the distance of the man from the building as well as the distance between him and the top of the tower.

C(man)

B

A

45˚

30 m

Distance (BC) tan45˚ = 1 = AB/BC = 30/BC BC = 30 mTherefore, the distance between the man and the tower is 30 meters.

Now, Finding AC sin45˚ = 1/√2 = 30/AC AC = 30 √2 metersThus, the distance between the man and the top of the tower is 30 √2 meters.

A man in a car is looking at the top of a tree, which is 40 m from him. Find the distance between the man and the top of the tree, if the angle of elevation is 30 degrees.

cos30˚ = √3 / 2 = 40 / AC AC = 80 / √3 Therefore, distance between the man and the top of the tree is 80 / √3 meters.

C(car) B

30˚

40 m

A