Maths Formulas

Algebra

Basic Identies

(a+b) 2 = a 2 + 2ab + b 2
(a-b) 2 = a 2 - 2ab + b 2
a 2 + b 2 = (a+b) 2 - 2ab
a 2 + b 2 = (a-b) 2 + 2ab
a 2 - b 2 = (a+b)(a-b) 
(a+b) 3 = a 3 + 3a2b + 3ab2 + b 3
(a-b) 3 = a 3 - 3a2b + 3ab2 - b 3
a 3 + b 3 = (a + b)(a2 - ab + b2) 
a 3 - b 3 = (a - b)(a2 + ab + b2)
(a+b)(c+d) = ac + ad + bc + bd
x 2 + (a+b)x + ab = (x + a)(x + b)

Powers

x a x b = x (a + b)
x a y a = (xy) a
(x a) b = x (ab)
x (-a) = 1 / x a
x (a - b) = x a / x b

Logarithms

y = logb(x) if and only if x=b y
logb(1) = 0
logb(b) = 1
logb(x*y) = logb(x) + logb(y)
logb(x/y) = logb(x) - logb(y)
logb(x n) = n logb(x)
logb(x) = logb(c) * logc(x) = logc(x) / logc(b)

Modern Algebra

Closure Property of Addition
Sum (or difference) of 2 real numbers equals a real number
Additive Identity
a + 0 = a
Additive Inverse
a + (-a) = 0
Associative of Addition
(a + b) + c = a + (b + c)
Commutative of Addition
a + b = b + a
Definition of Subtraction
a - b = a + (-b)
Trignometric Formulae
Relation Between Degree and Radians:
Angles in Degress
0
30
45
60
90
Angles in radians
0       
π / 6
π / 4
π / 3
π / 2


 1c = (180 / π)o

1o =  (π / 180)c

Trigonometric Ratios of an acute angle of a right triangle:

Sin θ = Length of opposite side / Length of hypotenuse side
Cos θ = Length of adjacent side / Length of hypotenuse side
Tan θ = Length of opposite side / Length of adjacent side
Sec θ = Length of hypotenuse side / Length of adjacent side
Cosec θ = Length of hypotenuse side / Length of opposite side
Cot θ = Length of adjacent side  / Length of opposite side 

Reciprocal Relations:

Sin θ = 1 / Cosec θ                       Sec θ = 1 / Cos θ

Cos θ = 1 / Sec θ                          Cosec θ = 1 / Sin θ

Tan θ = 1 / Cot θ                           Cot θ = 1 / Tan θ
  
Quotient Relations:

Tan θ = Sin θ / Cos θ        

Cot θ = Cos θ / Sin θ



Trigonometric ratios of Complementary angles:

Sin (90 – θ) = Cos θ                          Sec (90 – θ) = Cosec θ

Cos (90 – θ) = Sin θ                          Cosec (90 – θ) = Sec θ

Tan (90 – θ) = Cot θ                           Cot (90 – θ) = Tan θ

Trigonometric ratios for angle of measure:

  θ
  0
  30
 45
  60
  90
Sin θ
0
1/2
1/√2
√3/2
1
Cos θ
1
√3/2
1/√2
1/2
0
Tan θ
0
1/√3
1
√3
∞
Cot θ
∞
√3
1
1/√3
0
Sec θ
1
2/√3
√2
2
∞
Cosec θ
∞
2
√2
2/√3
1
  

Basic Identities

  • Sin 2 x + Cos 2 x = 1
  • Sec 2 x - Tan2 x = 1
  • Cosec 2 x - Cot 2 x = 1
  • Sin 2 x = 1 - Cos 2 x
  • Cos 2 x = 1 - Sin 2 x
  • Sec 2 x = 1 + Tan2x
  • Cosec 2 x = 1 + Cot2 x
  • Sec 2 x -1 = Tan2 x
  • Cosec 2 x –1 = Cot2 x

Addition and Difference 


  • Sin(A+B) = SinA CosB + CosA SinB
  • Sin(A-B) = SinA CosB - CosA SinB
  • Cos(A+B) = CosA CosB - SinA SinB
  • Cos(A-B) = CosA CosB + SinA SinB
  • Tan(A+B) = (TanA + TanB) / (1- TanA TanB)
  • Tan(A-B) = (TanA - TanB) / (1+ TanA TanB)
  • Sin2A = 2 SinA CosA
  • Cos2A = Cos2 A - Sin2 A = 2 Cos2 A - 1 = 1 - 2 Sin2 A
  • Tan2A = 2 Tan A / (1 - Tan2 A)
  • Sin2 A = (1 - Cos2A)/2
  • Cos2 A = (1 + Cos2A)/2
  • Sin2A = 2TanA/ (1+Tan2A)
  • Cos2A = (1-Tan2A)/ (1 + Tan2A)
  • Tan2A = 2TanA/ (1-Tan2A)
  • Sin A = 2 SinA/2 CosA/2 = 2TanA/2 / (1+Tan2A/2)
  • Cos A = Cos2 A/2 - Sin2 A/2 = 2 Cos2 A/2 - 1 = 1 - 2 Sin2 A/2 = (1-Tan2A/2)/ (1 + Tan2A/2)
  • TanA = 2TanA/2 / (1-Tan2A/2)
  • Sin 3A = 3 Sin A - 4 Sin3A
  • Cos 3A = 4 Cos3A- 3 Cos A
  • Tan 3A = (3Tan A – tan3A) / (1-3Tan2A)
  • Cos2A/2 = 1 + Cos A/2
  • Sin2A/2 = 1 - Cos A/2
  • Cos3A = (3Cos A + Cos 3A)/4
  • Sin3A = (3Sin A + Sin 3A)/4
  • Sin 18 = (√5 – 1)/ 4
  • Cos 36 = (√5 + 1)/ 4
  • Sin(A+B) Sin(A-B) = Sin2 A - Sin2 B
  • Cos(A+B) Cos(A-B) = Cos2 A - Sin2 B
  • 2 Sin A Cos B = Sin(A+B) + Sin(A-B)
  • 2 Cos A Sin B = Sin(A+B) - Sin(A-B)
  • 2 Cos A Cos B = Cos(A+B) + Cos(A-B)
  • 2 Sin A Sin B = Cos(A-B) - Cos(A+B)
  • Sin C - Sin D = 2 Sin( (C - D)/2 ) Cos( (C + D)/2 )
  • Cos C - Cos D = -2 Sin( (C - D)/2 ) Sin( (C + D)/2 )
  • Sin C + Sin D = 2 Sin( (C + D)/2 ) Cos( (C - D)/2 )
  • Cos C + Cos D = 2 Cos( (C - D)/2 ) Cos( (C + D)/2 )
   

Hyperbolic Definitions
sinh(x) = ( e x - e -x )/2
cosech(x) = 1/sinh(x) = 2/( e x - e -x )
cosh(x) = ( e x + e -x )/2
sech(x) = 1/cosh(x) = 2/( e x + e -x )
tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x )
coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x )
cosh 2(x) - sinh 2(x) = 1
tanh 2(x) + sech 2(x) = 1
coth 2(x) - cosech 2(x) = 1


Inverse Hyperbolic Definitions
Sinh-1(z) = log( z + (√z2 + 1) )
Cosh-1(z) = log( z + (√z2 - 1) )
Tanh-1(z) = 1/2 log( (1+z)/(1-z) )



Relations to Trigonometric Functions
sinh(z) = -i sin(iz)
cosech(z) = i cosec(iz)
cosh(z) = cos(iz)
sech(z) = sec(iz)
tanh(z) = -i tan(iz)
coth(z) = i cot(iz)
sin(-x) = -sin(x)
cosec(-x) = -cosec(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)



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