MATHEMATICS ENCYCLOPEDIA

A theodolite ( Application of trigonometry) A theodolite is a instrument used for measuring angles in vertical and horizontal planes. A theodolite is used for the applications of trigonometry such as surveying, navigation, measuring distances etc. A theodolite consist of movable telescope mounted on two perpendicular axes. Surveyors have used trigonometry for centuries. One such large surveying project of the nineteenth century was the 'Great Trigonometry Survey' of British India for which the two largest-ever theodolites were built . During the survey in 1852, the highest mountain in the world was discovered. From a distance of over 160 km , the peak was observed from six different stations. In 1856, this peak was named after Sir George Everest, who had commissioned and first used the giant theodolites . The theodolites are now on display in the Museum of the survey of India in Dehradun .

Measuring heights and distances using trigonometry. Trigonometry is also used for finding the heights and distances of various objects, without actually measuring them. To find the distance or height of any object or building etc., you have to calculate the angle of elevation/depression and calculate the distance/height using trigonometric ratios. Some important expressions to keep in mind while solving the above problem of height and distance given below: Line of sight: It is the line drawn from the eye of an observer to the point in the object viewed by the observer. Angle of elevation: This is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e., the case when we raise our head to look at the object. Angle of depression: The angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e., the case when we lower our head to look at the object.  Trigonometric ...

Frustum is a Latin word meaning 'piece cut off', and its plural is 'frusta'. A frustum is a portion of a solid that lie between one or two parallel planes. The frustum of cone has two circular ends with different radii. Take a right circular cone and remove a portion of it. There are many ways in which we can do this. But one particular case that we are interested in it is the removal of a smaller right circular cone by cutting the given cone by a plane parallel to its base. You must have observed that the glasses ( tumblers ) , in general, used for drinking water, are of this shape. Construction of frustum of cone Take some clay, or any other such material ( like plasticine, etc. ) and form a cone.  Cut it with a knife parallel to its base.  Remove the cone that is formed on one side of that plane. The part that now left is a frustum of cone. The frustum of cone has two circular ends with different radii. Let h be the height, l the slant...

Graph of function  f: R→R by y = f(x) =  x 2  

Trigonometric ratios of various angles The values of cosec x, sec x, cot x, are the reciprocal of the values of sin x, cos x, and tan x respectively. 

Angle is a measure of rotation of a given ray about its initial point.  The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlock, the angle is said to be positive . If the direction of rotation is clock, then the angle is negative . The measure of an angle is the amount of rotation performed to get the terminal side from the initial side. There are several units for measuring angles. Degree measure If a rotation from the initial side to terminal side is ( 1/360 )th of a revolution, the angle is said to have a measure of one degree ( 1° ). A degree is divided into 60 minutes ( 60' ). A minute is divided into 60 seconds ( 60" ). One sixtieth of a degree is called a minute, i.e. 1', and one sixtieth of a minute is called a second, i.e. 1". Thus,     1° = 60' ,  1' = 60...

Solution of triangles and trigonometry applications. In a triangle ABC , the angles are denoted by the capital letters A, B, C and the lengths of sides opposite to these angles are denoted by a, b, c respectively . See the figure. Some important formulae regarding the sides and angles of a triangle are given as follows: The law of  sines:  In any triangle ABC where  R  is the  radius of the circumcircle  of the triangle ABC. R is also known as the  circumradius  of the triangle. The law of  cosines:  In any triangle ABC Trigonometric ratios of half-angles in terms of the sides Let 2s = a + b + c, so that s is the semi-perimeter of triangle ABC. Then, Napier's anology: In any triangle ABC