mathematics is fun: November 2016

Wednesday 23 November 2016

AREA FORMULA

AREA OF PLANE FIGURE

DEFINITION:

The area of an object is the space occupied by it on a plane surface.

FORMULAE OF PLANE FIGURE:

1)AREA OF RECTANGLE=( l*b ) sq. units

where l is length and

b is breadth

2)AREA OF SQUARE= (a*a) sq. units

where a is the side

3)AREA OF RIGHT TRIANGLE= 1/2*(b*h) sq. units

where b is base and

h is height

4)AREA OF QUADRILATERAL=1/2*d*(h1*h2) sq. units

where d is the length of a diagonal and

h1 and h2 are perpendiculars drawn to the diagonal from the opposite vertives

5)AREA OF PARALLELOGRAM=bh sq. units

where b is the base and

h is the height

6)AREA OF RHOMBUS=1/2*(d1*d2) sq. units

where d1 and d2 are diagonals

7)AREA OF TRAPEZIUM=1/2*h(a+b) sq.units

where h is height and

a and b are sum of the parallel sides

8)AREA OF CIRCLE= $π(r*r)sq. units$

$where r is the radius of the semi-circle$

$9)AREA OF SEMICIRCLE=$ $π(r*r)/2 sq. units$

$where r is the radius of the circle$

$10)AREA OF A QUADRANT=1/4*$ $π*(r*r) sq. units$

$where r is the radius$

Thursday 17 November 2016

TRIGNOMETRIC FUNCTION:

1. Sine function(sin), defined as the ratio of the side of the opposite the angle to the hypotenuse.

$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$

2. Cosine function(cos), defined as the ratio of the adjacent leg to the hypotenuse.

$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$

3. Tangent function(tan), defined as the ratio of the opposite leg to the adjacent leg.

$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{a}{\,c\,}*\frac{c}{\,b\,}=\frac{a}{\,c\,} / \frac{b}{\,c\,}=\frac{\sin A}{\cos A}\,.$

The hypotenuse is the side opposite to the 90 degree angle in a right triangle, it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent side is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A.

The reciprocals of these functions are named the cosecant(cosec), secant(sec) and cotangent(cot).

$\csc A={\frac {1}{\sin A}}={\frac {{\textrm {hypotenuse}}}{{\textrm {opposite}}}}={\frac {c}{a}},$

$\sec A={\frac {1}{\cos A}}={\frac {{\textrm {hypotenuse}}}{{\textrm {adjacent}}}}={\frac {c}{b}},$

$\cot A={\frac {1}{\tan A}}={\frac {{\textrm {adjacent}}}{{\textrm {opposite}}}}={\frac {\cos A}{\sin A}}={\frac {b}{a}}.$

USES OF TRIGNOMETRY:

The technique of triangulation is used in astronomy to meausre the distance to nearby stars.
In geography to measure distance between landmarks.
In satellite navigation system.
The sine and cosine functions are fundamental to the theory of periodic functions, such as those that describe sound and light waves.
Fields that use trignometric functions include music theory, audio synthesis, accoustics, optics, electronics, biology, pharmacy, chemistry, number theory, electrical engineering and game development.

PYTHAGOREAN IDENTITIES:

The identities are related to the Pythagorean theorem and hold for any values.

   $\sin^2 A + \cos^2 A = 1 \$
   $\tan ^{2}A+1=\sec ^{2}A\$
   $\cot ^{2}A+1=\csc ^{2}A\$

ANGLE TRANSFORMATION FORMULAE:

   $\sin (A \pm B) = \sin A \ \cos B \pm \cos A \ \sin B$
   $\cos (A \pm B) = \cos A \ \cos B \mp \sin A \ \sin B$
   $\tan(A\pm B)={\frac {\tan A\pm \tan B}{1\mp \tan A\ \tan B}}$
   $\cot (A \pm B) = \frac{ \cot A \ \cot B \mp 1}{ \cot B \pm \cot A }$

Thursday 10 November 2016

Mathematicians of India

GREAT MATHEMATICIANS OF INDIA

ARYABHATA:

Aryabhata was born in 476AD. He called himself a native of Kusumapura or Pataliputra(Patna). He was died on 550AD. His achievements in mathematics are place value system and zero, approximation of π, trigonometry , indeterminate equations algebra.

BRAHMAGUPTA:

Brahmagupta was born in 598CE in Bhinmal, a state of Rajasthan, in India. He was died between 660 to 670. His achievements in mathematics are linear equation, arthimaetic, sum of squares and cubes of first n integer, mentions zero as a number, Pythagorean triples, pell’s equation, brahmagupta formula for cyclic quadrilaterals, Brahmagupta’s theorem, sine table and interpolation formula.

SRINIVASA RAMANUJAN:

Srinivasa Ramanujan was born in 22nd december1887 in Erode, Madras Presidency. He was died on 26^th April 1920. His achievements in mathematics are Ramanujan stato-series, Ramanujan conjecture and Ramanujan’s notebooks.