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Application Problems in Trigonometry Solution 1 The angle of elevation of the top of the building at a distance of 50m from its foot on a horizontal plane is found to be 60 degree. Solution 2 A ladder placed against a wall such that it reaches the top of the wall of height 6 m and the ladder is inclined at an angle of 60 degree.
Solution 3 From the top of the tower 30m height a man is observing the base of a tree at an angle of depression measuring 30 degree. Solution 4 A man wants to determine the height of a light house. Solution 5 The angle of elevation of the top of the light house at a distance of 30 m from its foot on a horizontal plane is found to be 60 degree.
Solution 7 Two men are on the opposite sides of a building. Solution 8 The angle of elevation of a multistoreyed building from a point on the toad changes from 30 degree to 60 degree as one walks m along the road towards the building,find the height of the building. Solution 9 A flag staff stands on the top of 6 m high tower.
Solution 10 The angles of depression of the top and the bottom of a 12 m high building from the top of the tower are 45 degree and 60 degree respectively. Calculate the height of the tower Solution. Recent Articles. The following table summarizes the properties of the graphs of the six main trigonometric functions:  . Because the six main trigonometric functions are periodic, they are not injective or, 1 to 1 , and thus are not invertible.
By restricting the domain of a trigonometric function, however, they can be made invertible. The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:  : 48ff  : ff. When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series.
For instance, sine and cosine have the following representations: . With these definitions the trigonometric functions can be defined for complex numbers.
Trigonometry may not have its direct applications in solving practical issues, but it is used in various things that we enjoy so much. For example. The aim of the unit is to both convince the students that trigonometry can be used for practical purposes, and consolidate the understandings they already have.
This complex exponential function, written in terms of trigonometric functions, is particularly useful. Trigonometric functions were among the earliest uses for mathematical tables. Scientific calculators have buttons for calculating the main trigonometric functions sin, cos, tan, and sometimes cis and their inverses. Most computer programming languages provide function libraries that include the trigonometric functions. In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today.
See List of trigonometric identities for more relations between these functions.
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,  predicting eclipses, and describing the orbits of the planets. In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,  as well as in satellite navigation systems. Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation. Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.
In land surveying , trigonometry is used in the calculation of lengths, areas, and relative angles between objects. On a larger scale, trigonometry is used in geography to measure distances between landmarks, . The sine and cosine functions are fundamental to the theory of periodic functions ,  such as those that describe sound and light waves. Fourier discovered that every continuous , periodic function could be described as an infinite sum of trigonometric functions.
Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics  and communications  , among other fields. Trigonometry is useful in many physical sciences ,  including acoustics ,  and optics . In these areas, they are used to describe sound and light waves , and to solve boundary- and transmission-related problems.
Other fields that use trigonometry or trigonometric functions include music theory ,  geodesy , audio synthesis ,  architecture ,  electronics ,  biology ,  medical imaging CT scans and ultrasound ,  chemistry ,  number theory and hence cryptology ,  seismology ,  meteorology ,  oceanography ,  image compression ,  phonetics ,  economics ,  electrical engineering , mechanical engineering , civil engineering ,  computer graphics ,  cartography ,  crystallography  and game development.
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities ,  relate both the sides and angles of a given triangle.
https://ciagranawthou.tk In the following identities, A , B and C are the angles of a triangle and a , b and c are the lengths of sides of the triangle opposite the respective angles as shown in the diagram. The law of sines also known as the "sine rule" for an arbitrary triangle states: . The law of cosines known as the cosine formula, or the "cos rule" is an extension of the Pythagorean theorem to arbitrary triangles: .
Given two sides a and b and the angle between the sides C , the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides: . Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a , b , and c , and if the semiperimeter is. The following trigonometric identities are related to the Pythagorean theorem and hold for any value: . Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.
From Wikipedia, the free encyclopedia. For other uses, see Trig disambiguation. In geometry, study of the relationship between angles and lengths. Main article: History of trigonometry. Main article: Trigonometric function. Main article: Mnemonics in trigonometry. Main article: Unit circle. Main article: Inverse trigonometric functions.
Main article: Trigonometric tables. Main article: Uses of trigonometry. Main article: Astronomy. Main article: Navigation. Main article: Surveying. Main articles: Fourier series and Fourier transform. Their summation is called a Fourier series. Main articles: Optics and Acoustics.
Main article: List of trigonometric identities. Online Etymology Dictionary. Nagel ed. A History of Mathematics 2nd ed.
A treatise on trigonometry, plane and spherical: with its application to navigation and surveying, nautical and practical astronomy and geodesy, with logarithmic, trigonometrical, and nautical tables. Trigonometry For Dummies. Halmos 1 December I Want to be a Mathematician: An Automathography. Hostetler 10 March Cengage Learning. A History of Mathematics. Episodes from the Early History of Astronomy.
New York: Springer.
A history of ancient mathematical astronomy. Book Sales.
MacTutor History of Mathematics archive. Retrieved One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications.
Applications of Fourier Transforms to Generalized Functions. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry. What does this mean? By restricting the domain of a trigonometric function, however, they can be made invertible. Trigonometric functions were among the earliest uses for mathematical tables. For other uses, see Trig disambiguation.
In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth. October Biography — Encyclopaedia Iranica".
His major contribution in mathematics Nasr, , pp. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles. Lennart Princeton University Press. Wilson From Byzantium to Italy. Greek Studies in the Italian Renaissance , London. Krebs Greenwood Publishing Group. From Kant to Hilbert: a source book in the foundations of mathematics. Oxford University Press US. Focus on Curves and Surfaces.