By AKSHAT GOYAL
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
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
History of Trigonometry
Classical 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
The first trigonometric
table was apparently
compiled by Hipparchus,
who is now consequently
known as "the father of
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.
Applications of Trigonometry
There 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
AngLe of depression
Angle of Depression t
Angle of Elevation
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.
A man is standing at a distance from a building of
height 30 m. The angle of elevation from the man’s
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. A
tan45˚ = 1 = AB/BC = 30/BC
BC = 30 m
Therefore, the distance between the man and the
tower is 30 meters.
Now, Finding AC
sin45˚ = 1/√2 = 30/AC
AC = 30 √2 meters
Thus, 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
cos30˚ = √3 / 2 = 40 / AC
AC = 80 / √3
Therefore, distance between the man and the top
of the tree is 80 / √3 meters.