SUBMITTED BY:

Kanishk Gupta

10th B

ACKNOWLEDGEMENTACKNOWLEDGEMENT• In the course of present work it has been our

privilege to receive help and assistance from many quarters. We take great pleasure. In acknowledging here, our debt to them.

• This endeavor has been possible with the kind and generous direction and encouragement given by Mr.Adhikari. We would like to show our gratitude to him.

• With the help of internet it has become easier to me to collect data.

CONTENTS1. Introduction2. An old surveyor's tele3. Great Trigonometric Survey4. Indian mathematics5. Varahamihira used the formulas6. SOME BASIC FORMULAS

INTRODUCTIONThe basic problem of

trigonometry runs somewhat like this:

• You stand next to a wide river and need to know the distance across it--say to a tree on the other shore, marked on the drawing here by the letter C (for simplicity, let's ignore the 3rd dimension).

How can this be done without actually crossing the river?

•The usual prescription is as follows. Stick two poles in the ground at points A and B, and with a tape measure or surveyor's chain measure the distance c between them ("the baseline").

An old surveyor's telescope (theodolite)

Then remove the pole at A and replace it with a surveyor's telescope like the one shown here ("theodolite"), having a plate divided into 360 degrees, marking the direction ("azimuth) in which the telescope is pointing. Sighting the telescope, first at the tree and then at pole B, you measure the angle A of the triangle ABC, equal to the difference between the numbers you have read from the azimuth plate.. Replace the pole, take your scope to point B and measure the angle B in the same way.

• The length c of the baseline, and the two angles A and B, contain all there is to know about the triangle ABC--enough, for instance, to construct a triangle of the same size and shape on some convenient open field. Trigonometry (trigon = triangle) was originally the art of deriving the missing information by pure calculation. Given enough information to define a triangle, trigonometry lets you calculate its remaining dimensions and angles.

• Why triangles? Because they are the basic building blocks from which any shape (with straight boundaries) can be constructed. A square, pentagon or another polygon can be divided into triangles, say by straight lines radiating from one corner to all others.

• In mapping a country, surveyors divide it into triangles and mark each corner by a "benchmark", which nowadays is often a round brass plate set into the ground, with a dimple in its center, above which the surveyors place their rods and telescopes (George Washington did this sort of work as a teenager). After measuring a baseline--such as AB in the example of the river--the surveyor would measure (as described here) the angles it formed with lines to some point C, and use trigonometry to calculate the distances AC and BC. These can serve as baselines for 2 more triangles, each of which provides baselines for two more... and so on, more and more triangles until the entire country is covered by a grid involving only known distances. Later a secondary grid may be added, subdividing the bigger triangles and marking its points with iron stakes, providing additional known distances on which any maps and plans can be based.

"Great Trigonometric Survey"

• The length c of the baseline, and the two angles A and B, contain all there is to know about the triangle ABC--enough, for instance, to construct a triangle of the same size and shape on some convenient open field. Trigonometry (trigon = triangle) was originally the art of deriving the missing information by pure calculation.

Given enough information to define a triangle, trigonometry lets you calculate its remaining dimensions and angles.

•Why triangles? Because they are the basic building blocks from which any shape (with straight boundaries) can be constructed. A square, pentagon or another polygon can be divided into triangles, say by straight lines radiating from one corner to all others.

•In mapping a country, surveyors divide it into triangles and mark each corner by a "benchmark", which nowadays is often a round brass plate set into the ground, with a dimple in its center, above which the surveyors place their rods and telescopes (George Washington did this sort of work as a teenager).

After measuring a baseline--such as AB in the example of the river--the surveyor would measure (as described here) the angles it formed with lines to some point C, and use trigonometry to calculate the distances AC abd BC. These can serve as baselines for 2 more triangles, each of which provides baselines for two more... and so on, more and more triangles until the entire country is covered by a grid involving only known distances. Later a secondary grid may be added, subdividing the bigger triangles and marking its points with iron stakes, providing additional known distances on which any maps and plans can be based.

Indian mathematics• Statue of Aryabhata.

As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.

A medieval artist's rendition of Claudius Ptolemy

– The next significant developments of trigonometry were in India. Influential works from the 4th–5th century, known as the Siddhantas (of which there were five, the most complete survivor of which is the Surya Siddhanta[13]) 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.[14] Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya.[15] The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.[16] They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.

Varahamihira used the formulas

• trigonometric equivalent to formulas known by Thales and Pythagoras

• modern sine and cosine equivalent to a chord formula known to Ptolemy; see above

• In the 7th century, Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(x), which had a relative error of less than 1.9%:

• Later in the 7th century, Brahmagupta redeveloped the formula (also derived earlier, as mentioned above) as well as the Brahmagupta interpolation formula for computing sine values.[18] Another later Indian author on trigonometry was Bhaskara II in the 12th century.

• Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the power series expansions of sine, cosine, tangent, and arctangent. Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the θ, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.

SOME BASIC FORMULAS