MATHEMATICS ENCYCLOPEDIA

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...

The number 111 111 is three times the prime number 37, and it the sum of the numbers from 16 to 21, see below: 111 = 3 * 37        = 16 + 17 + 18 + 19 + 20 + 21 In cricket, a score of 111 is called ‘The Nelson’ and is considered unlucky. A ‘Double Nelson’ is 222 .  Numbers like 111 which are just written with ones are called repunits .  See what happens when you square repunits. The pattern is below which goes on forever. 1 2 = 1 11 2 = 121 111 2 = 12321 1111 2 = 1234321 11111 2 = 123454321 111111 2 = 12345654321 1111111 2 = 1234567654321 11111111 2 = 123456787654321 111111111 2 = 12345678987654321

A woman was travelling with her young son, a carton of cakes and a wolf She had to cross a river in a small boat large enough for herself and one of the three . The following problems arises: If she left her son alone with the cakes, he could eat them.  If she left the wolf alone with the boy, the wolf would eat him. The question is how could she make the crossing ? Puzzled! See the solution below. Solution: The woman takes the boy across, returns alone. She takes the carton of cakes across, returns with a boy. She takes wolf across, return alone. Then she takes the boy across.

In A.D. 1120, the king of England decreed that in his country, the yard will be used as the unit of length. It was the distance from the tip of the king's nose to the end of his outstretched arm. The original standard adopted by France was called a foot . It was equal to the royal foot of the king. In 1799, the kings foot was replaced by "meter". Then it was defined as one ten millionth distance from the equator to the north pole along a longitudinal line that passes through Paris . In 1889, 1 meter was defined as the distance between two lines on a specific platinum-iridium bar stored under controlled conditions. In 1960, meter was redefined as 1,650,763.73 wavelengths of a particular orange-red light emitted by atoms of Krypton-86 in a gas discharge tube . 1 meter = 100 cm The meter was redefined again in 1983, this time as the distance which light travel in a vacuum in 1/299,792,458 seconds. This remains the current definition. Note that in all these re...