more on maths   Awaiting Review for Nickels Tagged with:  academic    posted on 20 Jul 2008 08:48:13 IST
38 More Math Formulae, Tricks and Shortcuts     If n is even , n(n+1)(n+2) is divisible by 24 If n is any integer , n^2 + 4 is not divisible by 4   Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for [(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]   Area of a triangle 1/2*base*altitude = 1/2*a*b*sinC = 1/2*b*c*sinA = 1/2*c*a*sinB = root(s*(s-a)*(s-b)*(s-c)) where s=a+b+c/2 =a*b*c/(4*R) where R is the CIRCUMRADIUS of the triangle = r*s ,where r is the inradius of the triangle . In any triangle a=b*CosC + c*CosB b=c*CosA + a*CosC c=a*CosB + b*CosA   If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to (k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to (a1+a2+a3+............./b1+b2+b3+..........)   (7)In any triangle a/SinA = b/SinB =c/SinC=2R , where R is the circumradius   x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3)   e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity 2 < e < 3   log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [ Note the alternating sign . .Also note that the logarithm is with respect to base e ]   In a GP the product of any two terms equidistant from a term is always constant.   For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2   For a cyclic quadrilateral, the measure of an external angle is equal to the measure of the internal opposite angle. (m+n)! is divisible by m! * n! .   If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal to the sum of the other pair .   The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP . The equation whose roots are the reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a   The coordinates of the centroid of a triangle with vertices (a,b) (c,d) (e,f) is((a+c+e)/3 , (b+d+f)/3) .   The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .   Area of a parallelogram = base * height   APPOLLONIUS THEOREM: In a triangle , if AD be the median to the side BC , then AB^2 + AC^2 = 2(AD^2 + BD^2) or 2(AD^2 + DC^2) .   For similar cones, ratio of radii = ratio of their bases. The HCF and LCM of two nos. are equal when they are equal.   Volume of a pyramid = 1/3 * base area * height   In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.   In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.   The quadrilateral formed by joining the angular bisectors of another quadrilateral is always a rectangle.   Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1   |a|+|b| = |a+b| if a*b>=0 else |a|+|b| >= |a+b|   2<= (1+1/n)^n <=3   WINE and WATER formula: If Q be the volume of a vessel q qty of a mixture of water and wine be removed each time from a mixture n be the number of times this operation be done and A be the final qty of wine in the mixture then , A/Q = (1-q/Q)^n   Area of a hexagon = root(3) * 3 * (side)^2   (1+x)^n ~ (1+nx) if x<<<1   Some Pythagorean triplets: 3,4,5 (3^2=4+5) 5,12,13 (5^2=12+13) 7,24,25 (7^2=24+25) 8,15,17 (8^2 / 2 = 15+17 ) 9,40,41 (9^2=40+41) 11,60,61 (11^2=60+61) 12,35,37 (12^2 / 2 = 35+37) 16,63,65 (16^2 /2 = 63+65) 20,21,29(EXCEPTION)   Apollonius theorem could be applied to the 4 triangles formed in a parallelogram.   Area of a trapezium = 1/2 * (sum of parallel sides) * height = median * height where median is the line joining the midpoints of the oblique sides.   When a three digit number is reversed and the difference of these two numbers is taken , the middle number is always 9 and the sum of the other two numbers is always 9 .   Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .   Let W be any point inside a rectangle ABCD . Then WD^2 + WB^2 = WC^2 + WA^2 Let a be the side of an equilateral triangle. Then if three circles be drawn inside this triangle touching each other then each has a radius = a/(2*(root(3)+1))  Let 'x' be certain base in which the representation of a number is 'abcd' , then the decimal value of this number is a*x^3 + b*x^2 + c*x + d When you multiply each side of the inequality by -1, you have to reverse the direction of the inequality.   To find the squares of numbers from 50 to 59 For 5X^2 , use the formulae   (5X)^2 = 5^2 +X / X^2   E.g. (55^2) = 25+5 /25 =3025 (56)^2 = 25+6/36 =3136 (59)^2 = 25+9/81 =3481   a+b+(ab/100) This is used for successive discounts types of sums. Like in 1999 population increases by 10% and then in 2000 by 5% So the population in 2000 now is 10+5+(50/100)=+15.5% more that was in 1999 and if there is a decrease then it will be preceded by a negetive sign and likewise.	About the Author: city_madhuri (73) Cool goIITian  13  [17 rates] total posts: 42     Offline
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comment by a4asd    (posted on 20 Jul 2008 09:45:05 IST) gr8
comment by digs2digi    (posted on 20 Jul 2008 11:39:33 IST) great work...

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