MATHEMATICS ENCYCLOPEDIA

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

Conic Sections These curves are known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone. These curves have a very wide range of applications in fields such as planetary motions , design of telescopes and antennas , reflectors in flashlights and automobile headlights etc. Sections of a cone Let l be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle α . Suppose we rotate the line m around the line l in such a way that the angle α remains constant.  Then the surface generated is a double-napped right circular hollow cone herein after referred to as cone and extending indefinitely far in both directions. The point V is called the vertex ; the line l is the axis of the cone.  The rotating line m is called the generator of the cone.  The vertex separates the cone into two parts called na...