list of formulae

LIST OF FORMULAE Length in metric units 1 m = 100 cm = 1000 mm Areas in metric units

1 m2  104 cm2 1 m2  106 mm2

1cm2  104 m2 1mm2  106 m2

1 cm2  102 mm2

1mm2  102 cm2

Volumes in metric units

1cm3  106 m3

1m3  106 cm2 1 litre = 1000 cm 3 1 m3  109 mm3

1mm3  109 m3 1 mm3  103 cm3

1 cm3  103 mm3

Laws of indices:

a m  a n = a m n m

an =

Quadratic formula:

a

n

= a m n

a n =

(a m ) n = a mn

1 an

a0 = 1

b  b2  4ac If ax + bx + c = 0 then x = 2a 2

Equation of a straight line:

Definition of a logarithm:

Laws of logarithms:

am

n

am

y = mx + c

If y = a x then x = log a y

log(A  B) = log A + log B

A log   = log A - log B B

log A n = n  log A

Exponential series:

ex = 1 + x +

x2 x3 + + .. 2! 3! (valid for all values of x)

Theorem of Pythagoras:

b2 = a2+ c2

Figure F1

Areas of plane figures: Area = l  b

(i) Rectangle

Figure F2

Area = b  h

(ii) Parallelogram (iii)Trapezium

Area =

(iv) Triangle

Area =

1 (a + b)h 2

1 bh 2

Area = r 2

(v) Circle

Radian measure:

Figure F3 Figure F4 Figure F5

Circumference = 2r

Figure F6

2 radians = 360 degrees

For a sector of circle: arc length, s =

shaded area =

 (2r) = r ( in rad) 360

 1 (r 2 ) = r 2  ( in rad) 360 2

Equation of a circle, centre at origin, radius r: x 2 + y 2 = r 2 Equation of a circle, centre at (a, b), radius r: (x - a) 2 + (y - b) 2 = r 2

Volumes and surface areas of regular solids: (i) Rectangular prism (or cuboid)

Volume = l  b  h

Figure F7

Surface area = 2(bh + hl + lb) (ii) Cylinder

Volume = r 2 h

Figure F8

Total surface area = 2rh + 2r 2 (iii)Pyramid

If area of base = A and perpendicular height = h then: Volume =

1 Ah 3

Figure F9

Total surface area = sum of areas of triangles forming sides + area of base (iv) Cone

Volume =

1 r 2 h 3

Curved surface area = rl (v) Sphere

Volume =

4 r 3 3

Figure F10 Total surface area = rl + r 2

Surface area = 4r 2

Figure F11

Areas of irregular figures by approximate methods: Trapezoidal rule

  widthof   1  first  last  Area       sumof remaining ordinates   int erval   2  ordinate   Mid-ordinate rule Area  (width of interval)(sum of mid-ordinates) Simpson’s rule Area 

1  widthof   first  last   sumof even   sumof remaining        4   2  3  int erval   ordinate   ordinates   oddordinates  

Mean or average value of a waveform mean value, y =

areaunder curve sumof mid  ordinates = length of base number of mid  ordinates

Triangle formulae: Sine rule:

a b c   sin A sin B sinC

Figure F12

Cosine rule: a 2 = b 2 + c 2 - 2bc cosA Area of any triangle =

=

1  base  perpendicular height 2

1 1 1 ab sin C or ac sin B or bc sin A 2 2 2

=

[s(s  a)(s  b)(s  c)] where s =

a b  c 2

For a general sinusoidal function y = A sin(t ± ), then A = amplitude

 = angular velocity = 2f rad/s

2 = periodic time T seconds 

 = frequency, f hertz 2

 = angle of lead or lag (compared with y = A sin t)

Cartesian and polar co-ordinates If co-ordinate (x, y) = (r, ) then r = x2  y 2 and  = arctan

y x

If co-ordinate (r, ) = (x, y) then x = r cos  and y = r sin 

Arithmetic progression: If a = first term and d = common difference, then the arithmetic progression is: a, a + d, a + 2d, .. The n’th term is : a + (n - 1)d Sum of n terms, S n =

n [2a + (n - 1)d] 2

Geometric progression: If a = first term and r = common ratio, then the geometric progression is: a, ar, ar 2 , .. The n’th term is: ar n 1 Sum of n terms, S n = If -1  r  1, S  =

a(1  r n) ar ( n  1) or (1  r) (r  1)

a (1  r)

Statistics: _

Discrete data: mean, x =

Grouped data:

x

_

mean, x =

n

fx f

standard deviation,  =

standard deviation,  =

_   2   (x  x)    n  

_    2   f(x  x)        f     

Standard derivatives y or f(x)

dy dx

or f’(x)

ax n

anx n 1

sin ax

a cos ax

cos ax

- a cos ax

e ax ln ax

ae ax 1 x

Standard integrals

 y dx

y

axn

a

xn  1 n 1

+ c (except when n = -1)

cos ax

1 sin ax + c a

sin ax

-

e ax 1 x

1 cos ax + c a 1 ax e +c a

ln x + c