Applications of Trigonometry :
Applications of Trigonometry Rishabh Tatiraju What is Trigonometry? :
What is Trigonometry? Trigonometry is a methodology for finding some unknown elements of a triangle provided the data includes a sufficient amount of linear and angular measurements to define a shape uniquely.
For example, two sides a and b of a triangle and the angle they include define the triangle uniquely. The third side c can then be found from the Law of Cosines while the angles α and β are determined from the Law of Sines.
Trigonometry is derived from Greek words trigōnon, meaning "triangle" and metron, meaning "measure“. History of Trigonometry :
History of Trigonometry The Babylonian astronomers (1900 BC) kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere. Some have even asserted that the ancient Babylonians had a table of secants.
The Egyptians used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. Slide 4:
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.
Soon afterwards, another Indian mathematician and astronomer, Aryabhata , collected and expanded upon the developments in his work, Aryabhatiya.
The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values. Applications of Trigonometry :
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 :
Line of Sight 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. Line of Sight Angle of Elevation :
Angle of Elevation 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. Angle of Elevation Angle of Depression :
Angle of Depression 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 Lets now solve some examples… :
Lets now solve some examples… Following are some very simple examples of the application of trigonometry.
In the first example, you have to find the distance of a man from the building as well as the distance between him and the top of the tower.
In the second example, you have to find Examples… :
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 Slide 11:
Distance (BC)
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. Slide 12:
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