these are the formulas we used so often in our mathematics classes...this is the ultimate collection of mathematical formulas right from 8 th std to 12 th
Trigonometric Formulas:
Cos (a+b) =cos a cos b-sin a sin b
Cos (a-b) =cos a cos b+ sin a sin b
Sin (a+b) =sin a cos b+ cos a sin b
Sin (a-b) =sin a cos b-cos a sin b
Analytical geometry formulas:
Plane:
Distance between two points of the slope, m=y2-y1/x2-x1
Circle:
Equation of a circle: (x-a)2+(y-b)2=r2
Centroid:
The centroid of a triangle vertices are P(x1,y1), P(x2,y2) and(x3,y3)
(x, y)=(x1+x2+x3/3, y1+y2+y3/3)
Parabola Formulas:
Y2=2ax
Parametric equation of the parabola:
X=2at2
y=2at
Hyperbola Formulas:
X2/a2-y2/b2=1
Parametric equation of the hyperbola:
X=a/sin t
Y=b sin t/cost
Ellipse formulas:
X2/a2+y2/b2=1
Parametric equation of the ellipse:
X=a cos t
Y=b sin t
Algebra formulas:
(a+b)2=a2+b2+2ab
(a-b)2=a2-b2+2ab
(a+b)3=a3+b3+3a2b+3ab2
(a-b)3=a3-b3-3a2b+3ab2
a2-b2=(a+b)(a-b)
a3-b3=(a-b)(a2+ab+b2)
a3+b3=(a+b)(a2-ab+b2)
(a+ b +c)2=a2+b2+c2+a(ab +bc +ca)
Temperature Scales:
Degrees Fahrenheit to Degrees Celsius:
TC = 5/9 (TF 32)
Degrees Celsius to Degrees Fahrenheit:
TF = 9/5 TC + 32
Degrees Celsius to Kelvin:
TK = TC + 273.15
Area:
Square:
A = a2, where a is one of the sides.
Rectangle:
A = ab, where a is the base and b is the height.
Trapezoid:
A = (h (a + b))/2, where h is a height, a is the longer parallel side, and b is the shorter.
Regular pentagon:
A = 1.720a2, where a is one of the sides.
Regular octagon:
A = 4.828a2, where a is one of the sides.
Circle:
A = r2, where r is the radius and 3.1416
Cube formulas:
Surface formulas:
Surface area of a cube=6a2
Surface area of a rectangle prism=2ab+2bc+2ca
. INTERGRATION
Formula List:
1. ∫ f(x) dx = f(x) + C , where c is an arbitrary constant.
2. ∫ [c1 f(x) + c2 g(x) ] dx = intc1 f(x) dx + int c2 g(x) dx
3. if K R then, ∫ k dx = kx + C
4. if n -1, then, ∫ xn dx = xn+1/n+1
5. if n = -1 then, ∫ xn = ∫ 1/ x dx = log |x| x C
6. ∫ ax = ax /log x + C
7. ∫ e x dx ∫ = ex + C
8. ∫ cos x dx = sinx + C
9. ∫ sinx dx = - cosx + C
10. ∫ sec2 x = tan x + C
11. ∫ cosec2x dx = - cot x+ C
12. ∫ secx tanx dx = sec x+ C
13. ∫cosecx cot x dx = - cosec x+ C
14. ∫ tanx dx = log | secx| + C
15. ∫ cot x dx = log | sinx| + C
16. ∫ secx dx = log | secx + tanx | + C
17. ∫ cosec x dx = log | cosecx -cotx | + C
2. ∫ [c1 f(x) + c2 g(x) ] dx = intc1 f(x) dx + int c2 g(x) dx
3. if K R then, ∫ k dx = kx + C
4. if n -1, then, ∫ xn dx = xn+1/n+1
5. if n = -1 then, ∫ xn = ∫ 1/ x dx = log |x| x C
6. ∫ ax = ax /log x + C
7. ∫ e x dx ∫ = ex + C
8. ∫ cos x dx = sinx + C
9. ∫ sinx dx = - cosx + C
10. ∫ sec2 x = tan x + C
11. ∫ cosec2x dx = - cot x+ C
12. ∫ secx tanx dx = sec x+ C
13. ∫cosecx cot x dx = - cosec x+ C
14. ∫ tanx dx = log | secx| + C
15. ∫ cot x dx = log | sinx| + C
16. ∫ secx dx = log | secx + tanx | + C
17. ∫ cosec x dx = log | cosecx -cotx | + C
Other Integral formulas list:
- ∫sinhx dx = cos hx + C
- ∫ cos hx dx = sinhx + C
- ∫ sec2 hx dx = tanhx + C
- ∫ sechx tanhx dx = - sechx + C
- ∫ cosechx cothx dx = -cosechx + C
- ∫ tanhx dx = log | cos hx | + C
- ∫ cothx dx = log +| sinhx | +C
Derivative:
Dx stands for derivative of the fn with respect to x
Constant function rule:
Dx a = 0, where ' a ' is a constant
Scalar multiple rule:
Dx (au) = a Dux, where ' a ' is a constant
Sum rule:
Dx (u + v) = Dux + Dvx
Difference rule:
Dx (u - v) = Dux - Dvx
Power rule:
Dx an = nan-1 Dux
Product rule:
Dx (uv) = u' v + uv'
Quotient rule:
Dx (u/v) = (u’v-v’u)/ v2
Reciprocal rule:
Dx (1/v)= (v’/v2)
Chain rule:
Dx f(g(x)) = f' (g(x)) g'(x)
These are the elementary rules of derivative.
List of Functional Derivative Rules:
List of trigonometric function rules:
Dx (sin x) = cos x
Dx (cos x) = - sin x
Dx (tan x) = sec2 x
Dx (cot x) = - cosec2x
Dx (sec x) = sec x tan x
Dx (cosec x) = - cosec x cot x
List of inverse trigonometric function rules:
Dx (sin-1 x) = 1/(1-x2)1/2
Dx (cos-1 x) = 1/(1-x2)1/2
Dx (tan-1 x) = 1/(1+x2)
Dx (sec-1 x) =1/ ( |x|(x2-1)1/2)
Dx (cosec-1 x) = 1/ ( |x|(x2-1)1/2)
Dx (cot-1 x) = - 1/(1 + x^2)
Exponential functions:
Dx (ev) = ev Dux
Dx (av) = av(ln a) Dux
Log functions:
Dx (log u) = 1/u Dux
List of hyperbolic functions:
Dx (sinh x) = cos hx
Dx (cosh x) = sin hx
Dx (tanh x) = sec h2x
Dx (sech x) = - tanh x sech x
Dx (cosech x) = - coth x cosech x
Dx (coth x) = - cosech2x
Binomial Formulas :
introduction toBinomial Formulas
· (x + y)n = xn + nxn-1y+n(n-1)/1.2 xn-2y2+...........+.nxyn-1+yn.
· (x - y)n = xn - nxn-1y+n(n-1)/1.2 xn-2y2+...........±.nxyn-1±yn.
MATHEMATICAL FORMULAS IN ALGEBRA:
1. N natural numbers formula = n (n+1)/2
2. Squares of first n natural numbers = n (n+1) (2n+1)/6
3. Cubes of first n natural numbers = [n (n+1)/2]2
4. Natural n number (odd) = n2
5. Average = (Sum of the items)/ Total Number of items.
Arithmetic Progression (A.P.): An A.P. is of the form a, a+d, a+2d, a+3d,
Where a is called the 'first term' and d is called the 'common difference'
Where a is called the 'first term' and d is called the 'common difference'
1. Nth term of an A.P is giev as , tn = a + (n-1) d
2. Sum of the first n terms of an A.P si gievn as, Sn = n/2[2a+ (n-1) d] or Sn = n/2(first term + Last term)
Geometrical Progression (G.P.): A G.P. is of the form a, a r, a r2, a r3,... Where a is called the 'first term' and r is called the 'common ratio'.
1. Nth term of a G.P is given as, tn = a rn-1
2. Sum of the first n terms in a G.P is given as, Sn = a|1-r n|/|1-r|
PERCENTAGES:
1. If A is R% more than B, then B is less than A by R / (100+R) * 100
2. If A is R% less than B, then B is more than A by R / (100-R) * 100
3. If the price of a commodity is increased by R%, then reduction in consumption, not To increase the expenditure is: R/(100+R)*100
4. If the price of a commodity is decreased by R%, then the increase in consumption, Not to decrease the expenditure is: R/(100-R)*100
PROFIT & LOSS:
1. Gain = Selling Price (S.P.) - Cost Price (C.P)
2. Loss = C.P. - S.P.
3. Gain percentage % = Gain * 100 / Cost of Price.
4. Loss percentage % = Loss * 100 / Cost of Price.
5. Selling Price = (100+Gain %) /100*C.P.
6. Selling Price = (100-Loss %) /100*C.P.
RATIO & PROPORTIONS:
1. The ratio a: b represents a fraction a/b. a is called as antecedent and b is called as consequent.
2. The equality of two different ratios is called proportion.
3. If a : b = c : d then a, b, c, d are in proportion. This is represented by a :b :: c : d.
4. In a : b = c : d, then we have a* d = b * c.
5. If a/b = c/d then ( a + b ) / ( a – b ) = ( d + c ) / ( d – c ).
AREA & PERIMETER:
1. Area of triangle formula = 1/2*Base*Height
2. Another Area of a triangle formula = square root of (s(s-(s-b)(s-c)) where a,b,c are the lengths of the sides and s= (a+b+c)/2
3. Area of parallelogram formula = Base * Height
4. Area of rhombus formula = 1/2(Product of diagonals)
5. Area of trapezium formula = 1/2(Sum of parallel sides)*(distance between the parallel sides)
6. Quadrilateral area formula = 1/2(diagonal)(Sum of sides)
7. Area for regular hexagon formula = 6(square root of 3/4)(side)2
8. Area for ring = square root of(R2-r2), where R is the outer radii and r is the inner radii of the ring.
Algebra formulas:
(a+b)2=a2+b2+2ab
(a-b)2=a2-b2+2ab
(a-b)3=a3-b3-3a2b+3ab2
(a+b)3=a3+b3+3a2b+3ab2
a2-b2=(a+b)(a-b)
a3-b3=(a-b)(a2+ab+b2)
a3+b3=(a+b)(a2-ab+b2)
(a+ b +c)2=a2+b2+c2+2(ab +bc +ca)
Geometry formulas:
Area:
Area of a Rectangle = b h
Area of Triangle = ½ b h
Area of a square = a2
Area of a Parallelogram = b × h
Area of a Circle = π r2
Volume:
Volume of cylinder = r2 h
Volume of cone = 1/3 π r2 h
Volume of cube = a3
Volume of Sphere = (4/3) π r3
Perimeter:
Perimeter of square = 4 a
Perimeter of rectangle = 2(l+b)
Perimeter of cube = 12 a
Circle:
Equation of a circle: (x-a)2+(y-b)2= r2
Centroid:
The centroid of a triangle with vertices P(x1,y1), P(x2,y2) and(x3,y3)
(x, y)=(x1+x2+x3/3, y1+y2+y3/3)
List of Trignometry and Line Formulas:
Line formulas:
Slope intercept form: y = mx + b
Midpoint formula = ((x1+x2)/2,(y1+y2)/2)
Point slope formula = (y – y1) = m (x- -x1)
Trigonometry formulas:
Sin A = opposite/hypotenuse
Cos A = adjacent/hypotenuse
Tan A = sin A/Cos A
Csc A = 1/Sin A
Sec A = 1/Cos A
Cot A = 1/Tan A
Sin2A + Cos2A = 1
Tan2A + 1 = Sec2A
1 + Cot2A = Csc2A
Formulae Used in Mathematical Series:
Arithmetic series:
The arithmetic series is a, a+ d, a+ 2d, a+ 3d . . . . . . .
The difference between the two consequent numbers in the series is d.
Formula used to find nth term is
Tn = a+ (n-1) d
Where n is the number of terms in the series.
Sum of the series
Sn =n(a1+an)/2
Geometric series:
The geometric series is a, ar, ar2, ar3 . . . . . . .
The ratio between two consequent numbers is r.
Formula used to find nth term is
Tn = arn-1 where n is the number of terms in the series.
Sum of the series
Sn = a(1-rn)/1-r
Sum or difference of two angles:
sin (a ± b ) = sin a cos b ± cos a sin b
cos(a ± b) = cos a cos b ± sin a sin b
tan(a ± b) =( tan a ± tanb)/1±tan a*tan b
Co-function identities:
Sin (π/2-ø) = cos ø
Cos (π/2-ø)= sin ø
Tan (π/2-ø) = cot ø
Csc (π/2-ø)= sec ø
Sec (π/2-ø)= csc ø
Cot (π/2-ø)= tan ø
Reduction formulas:
Sin (-ø) = - sin ø
Cos (-ø) = cos ø
Tan (-ø) = -tan ø
Csc (-ø) = csc ø
Sec (-ø) = sec ø
Cot (-ø) = - cot ø
Power reducing formulas:
Sin2 ø= (1-cos(2ø))/2
Cos2 ø= (1+cos(2ø))/2
Tan2 ø= (1-cos(2ø))/ (1+cos(2ø))
Sine rule:
Law of cosines:
a2 = b2 + c2 - 2bc A
Where A is the angle of a scalene triangle opposite side a
List of Integral formulas for hyperbolic trigonometric functions:
· ∫ 1/(a2+x2)dx = 1/a tan -1(x/a) + C
· ∫ 1/(x2- a2) dx = 1/2a log |(x-a)/(x+a) | + C
· ∫ f1(x) / f(x) dx = log | f(x) + C
· ∫[ f(x) ] n f1 (x) dx = [f(x)]n+1/n+1 + C
List of Elementary Rules of Derivative:
Constant function rule:
Dx a = 0, where ' a ' is a constant
Scalar multiple rule:
Dx (au) = a Dux, where ' a ' is a constant
Sum rule:
Dx (u + v) = Dux + Dvx
Difference rule:
Dx (u - v) = Dux - Dvx
Power rule:
Dx an = nan-1 Dux
Product rule:
Dx (uv) = u' v + uv'
Chain rule:
Dx f(g(x)) = f' (g(x)) g'(x)
ei x = cos( x ) + i sin( x )
sin x = x - x3/3! + x5/5! - x7/7! + x9/9! - x11/11! + ...
cos x = 1 - x2/2! + x4/4! - x6/6! + x8/8! - x10/10! + ...
ex = 1 + x + x2/2! + x3/3! + x4/4! + x5/5! + x6/6! + x7/7! + x8/8! + x9/9! + x10/10! + x11/11!
cos x = 1 - x2/2! + x4/4! - x6/6! + x8/8! - x10/10! + ...
ex = 1 + x + x2/2! + x3/3! + x4/4! + x5/5! + x6/6! + x7/7! + x8/8! + x9/9! + x10/10! + x11/11!
Integral Formula for Inverse Trigonometric Function:
The integral of inverse trigonometric sine function,
1. ∫ sin-1x dx = (1-x2)1/2 + x sin-1x + c
The integral of inverse trigonometric cosine function,
2. ∫cos-1x dx = x cos-1x - (1-x2)1/2 + c
The integral of inverse trigonometric tangent function,
3. ∫tan-1x dx = x tan-1x – (1/2) log (x2 + 1) + c
Other Trigonometric integral formulas:
4. ∫1/(a2+x2)dx = 1/a tan-1(x/a) + c
5. ∫ 1/(a2-x2)1/2dx= sin-1(x/a) + c
