maths formula for everyone final

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Maths Formula For Every One

Basic Algebra




Basic Trigonometry

Trigonometric Identities

sin(theta) = a / c cosesc(theta) = 1 / sin(theta) = c / a

cos(theta) = b / c sec(theta) = 1 / cos(theta) = c / b

tan(theta) = sin(theta) / cos(theta) = a / b cot(theta) = 1/ tan(theta) = b / a




sin(-x) = -sin(x) csc(-x) = -csc(x) cos(-x) = cos(x) sec(-x) = sec(x) tan(-x) = -tan(x) cot(-x) = -cot(x)

sin2(x) + cos2(x) = 1 tan2(x) + 1 = sec2(x) cot2(x) + 1 =

csc2(x)

sin(x y) = sin x cos y cos x sin y cos(x y) = cos x cosy sin x sin y

tan(x y) = (tan x tan y) / (1 tan x tan y)

sin(2x) = 2 sin x cos x

cos(2x) = cos2(x) - sin2(x) = 2 cos2(x) - 1 = 1 - 2 sin2(x)

tan(2x) = 2 tan(x) / (1 - tan2(x))

sin2(x) = 1/2 - 1/2 cos(2x)

cos2(x) = 1/2 + 1/2 cos(2x)

sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )

cos x - cos y = -2 sin( (x-y)/2 ) sin( (x + y)/2 )




Trig Table of Common Angles angle 0 30 45 60 90 sin2(a) 0/4 1/4 2/4 3/4 4/4 cos2(a) 4/4 3/4 2/4 1/4 0/4 tan2(a) 0/4 1/3 2/2 3/1 4/0

Given Triangle abc, with angles A, B,C; a is opposite to A, b opposites B, c opposite C:

a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)

c2 = a2 + b2 - 2ab cos(C)

b2 = a2 + c2 - 2ac cos(B)

a2 = b2 + c2 - 2bc cos(A)

(Law of Cosines)




Derivation

Definitions of the Derivative:

(right sided)

(left sided)

(both sided)

(Fundamental Theorem for Derivatives)

1. Constant Rule:

If y = k, then y' = 0 The Derivative of a Constant is 0

If (x) = k for some constant k, then '(x) = 0

2. Power Rule

If y = x", then y' = nxn-1

If is a differentiable function, and if (x) = x", then '(x) = nxn-1 for any real number n




3. Exponential Rule:

If y = ex, then y' = ex

4. Logarithm Rule:

If y = 1n|x|, then y' = 1/x

5. Constant Times a Function Rule:

If y = k, then y' = kf '

6. Sum Rule

If y = g, then y' = ' g'

7. Product Rule

If y = g, then y' = g' ' g

If and g are differentiable functions such that y = (x)g(x), then y' = (x)g' ' (x)g(x)

8. Difference Rule

If y = - g, then y' = ' - g'

9. Quotient Rule

To remember this formula: Simply remember that b comes before t in the alphabet Thus, the bottom function times the derivative of the top minus the top times the derivative of the bottom, all divided by the bottom squared!




10. Chain Rule:

If y is a differentiable function of u and u is a differentiable function of x and

(Fundamental Theorem for Derivatives)

c f(x) = c

f(x) (c is a constant)

(f(x) + g(x)) = f(x) + g(x)

f(g(x)) = f(g) * g(x) (chain rule)

f(x)g(x) = f' (x)g(x) + f(x)g '(x) (product rule)

(quotient rule)

Partial Differentiation Identities

if f( x(r,s), y(r,s) ) if f( x(r,s) )




Table of Derivatives

c = 0 x = 1 xn = n x(n-1)

ex = ex bx = bx ln(b) ln(x) = 1/x

sin x = cos x csc x = -csc x cot x

cos x = - sin x sec x = sec x tan x

tan x = sec2 x cot x = - cosec2 x

arcsin x = 1

(1 - x2)

arccsc x = -1

|x| (x2 - 1)

arccos x = -1

(1 - x2)

arcsec x = 1

|x| (x2 - 1)

arctan x = 1

1 + x2

arccot x = -1

1 + x2

sinh x = cosh x cosecsch x = - coth x cosech x

cosh x = sinh x sech x = - tanh x sech x

tanh x = 1 - tanh2 x coth x = 1 - coth2 x




First and Second Order Differential Equations

First Order Differential equations

A first order differential equation is of the form:

Linear Equations:

The general general solution is given by

where

is called the integrating factor.

Separable Equations:

(1) Solve the equation g(y) = 0 which gives the constant solutions. (2)




The non-constant solutions are given by

Bernoulli Equations:

(1)

Consider the new function . (2) The new equation satisfied by v is

(3) Solve the new linear equation to find v. (4)

Back to the old function y through the substitution . (5) If n > 1, add the solution y=0 to the ones you got in (4).

Homogenous Equations:

is homogeneous if the function f(x,y) is homogeneous, that is

By substitution, we consider the new function




The new differential equation satisfied by z is

which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by

Do not forget to go back to the old function y = xz.

Exact Equations:

is exact if

The condition of exactness insures the existence of a function F(x,y) such that

All the solutions are given by the implicit equation

Second Order Differential equations




Homogeneous Linear Equations with constant coefficients:

Write down the characteristic equation

(1)

If and are distinct real numbers (this happens if ), then the general solution is

(2)

If (which happens if ), then the general solution is

(3)

If and are complex numbers (which happens if ), then the general solution is

where

that is

Non Homogeneous Linear Equations:




The general solution is given by

where is a particular solution and is the general solution of the associated homogeneous equation

In order to find two major techniques were developed.

Method of undetermined coefficients or Guessing Method

This method works for the equation

where a, b, and c are constant and

where is a polynomial function with degree n. In this case, we have

where

The constants and have to be determined. The power s is equal to 0 if

is not a root of the characteristic equation. If is a simple root, then s=1 and s=2 if it is a double root. Remark. If the nonhomogeneous term g(x) satisfies the following




where are of the forms cited above, then we split the original equation into N equations

then find a particular solution . A particular solution to the original equation is given by

Method of Variation of Parameters

This method works as long as we know two linearly independent solutions

of the homogeneous equation

Note that this method works regardless if the coefficients are constant or not. a particular solution as

where and are functions. From this, the method got its name. The functions and are solutions to the system:

which implies




Therefore, we have

Euler-Cauchy Equations:

where b and c are constant numbers. By substitution, set

then the new equation satisfied by y(t) is

which is a second order differential equation with constant coefficients.

(1) Write down the characteristic equation

(2) If the roots and are distinct real numbers, then the general solution is given by

(2)If the roots and are equal ( ), then the general solution is




(3)If the roots and are complex numbers, then the general solution is

where and .

Integration

Integral Identities

Formal Integral Definition:

when...

a = x0 < x1 < x2 < ... < xn = b d = max (x1-x0, x2-x1, ... , xn - x(n-1)) xk-1




cos x dx = sin x + C cosesc x cot x dx = - csc x + C

sin x dx = -cos x + C sec x tan x dx = sec x + C

sec2 x dx = tan x + C csc2 x dx = - cot x + C

arcsin x dx = x arcsin x + (1-x2) + C

arccsc x dx = x arccos x - (1-x2) + C

arctan x dx = x arctan x - (1/2) ln(1+x2) + C

Useful Identities

arccos x = /2 - arcsin x (-1




Laplace Transforms

f(x) = e^(-xt) g(t) dt (Laplace Transform)

f(x) = e^(-xt) g(t) d (t) (Laplace-Stieltjes Transform)

f2(x) = L{L{g(t)}} = g(t)/(x+t) dt (Stieltjes Transform)

Elementary Laplace Transforms

=

(1)

=

(2)

=

(3)

=

(4)

=

(5)

=

(6)

=

(7)

=

(8)

=

(9)




=

(10)

=

(11)

=

(12)

=

(13)

=

(14)

= e-cs (15)

=

(16)

Fourier Transforms

f(x) = 1/ (2 ) g(t) e^(i tx) dt (Fourier Transform)

f(x) = (2/ ) g(x) cos(xt) dt (Cosine Transform)

f(x) = (2/ ) g(x) sin(xt) dt (Sine Transform)




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maths formula for everyone final

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