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## Maths Formulas

Sometimes, Math is Fun and sometimes it could be a surprising fact too. In our routine life, you can check the best route to your school, you can check where more discounted products are available in the market, and you can check which bank can offer the superior interests. This is all about calculation and connecting dots that we are able to find the solution.

#### Mensuration Maths Formulas

 Formula of Square Area of Square = $$l^{2}$$ Perimeter of Square = $$4 \times l$$ Where, l : length of side Formula of Rectangle Area of Rectangle = $$l \times w$$ Perimeter of Rectangle = $$2 (l+w)$$ Where, L = Length, w = Width Formula of Circle Area of Circle = $$\pi r^{2}$$ Perimeter of Circle = $$2 \pi r$$ Where, 𝒓 = Radius, d = Diameter, d = 2𝒓 Formula of Scalene Triangle Area of Scalene Triangle = $$\sqrt{s(s-a)(s-b)(s-c)}$$ Perimeter of Scalene Triangle = $$a+b+c$$ Where, a, b, c are Side of Scalene Triangle Formula of Isoscele Triangle Area of Isoscele Triangle = $$\frac{1}{2}bh$$ Altitude of an Isosceles Triangle = $$\sqrt{a^{2}-\frac{b^{2}}{4}}$$ Perimeter of Isosceles Triangle,P = $$2a+b$$ Where, b = Base, h = Height, a = length of the two equal sides Formula of Right Triangle Area of an Right Triangle = $$\frac{\sqrt{1}}{2}bh$$ Perimeter of an Right Triangle = $$a+b+c$$ semi Perimeter of an Right Triangle = $$\frac{a+b+c}{2}$$ where, b = Base, h = Hypotenuse, a = Hight Formula of Equilateral Triangle Area of an Equilateral Triangle = $$\frac{\sqrt{3}}{4}a^{2}$$ Perimeter of an Equilateral Triangle = $$3a$$ Semi Perimeter of an Equilateral Triangle = $$\frac{3a}{2}$$ Height of an Equilateral Triangle = $$\frac{\sqrt{3}}{2}a$$ Where, a = side, h = altitude Formula of Rhombus Perimeter of a rhombus = $$4\times Side$$ Area of a Rhombus A = $$\frac{1}{2} \times d_{1} \times d_{2}$$ Where, d1 and d2 are the diagonals Formula of Parallelogram Area of a Parallelogram = $$b\times h$$ Perimeter of Parallelogram = $$2\left(b+h\right)$$ Where, b: Base, h: Height. Formula of Trapezoid Perimeter of a Trapezoid = $$a+b+c+d$$ Area of a Trapezoid = $$\frac{1}{2} \times h \times (a + b)$$ Where: h = height, a = the short base, b = the long base, c, d are the lengths of side. Formula of Cube Surface area of Cube = $$6a^{2}$$ Volume of a cube = $$a^{3}$$ Where, a is the side length of the cube. Formula of Cuboid Surface area of Cuboid = $$2(lb + bh + hl)$$ Volume of a Cuboid = $$h \times l \times w$$ Where, l: Height, h: Legth, w: Depth Formula of sphere Diameter of a sphere = $$2r$$ Circumference of a sphere = $$2\pi r$$ Surface area of a sphere = $$4\pi r^{2}$$ Volume of a sphere = $$\frac{4}{3}\: \pi r^{3}$$ Where, r: Radius Formula of Hemisphere Curved Surface area of a Hemisphere = $$4\pi r^{2}$$ Total Surface area of a Hemisphere = $$3\pi r^{2}$$ Volume of a Hemisphere = $$\frac{2}{3}\: \pi r^{3}$$ Where, r: Radius Formula of Cylinder Curved Surface area of a Cylinder = $$2\pi rh$$ Total Surface area of a Cylinder = $$2\pi r(r+h)$$ Volume of a Cylinder = $$\pi r^{2} h$$ Where, r: Radius, h: Height Formula of cone Total Surface Area of cone = $$\pi r \left (s+r \right )$$ Vomule of cone = $$\frac {1}{3}\pi r^{2}h$$ Curved Surface Area of cone = $$\pi rs$$ Where, r = radius, h = height, s = slant height

#### Trigonometry Maths Formulas

• $$\sin \theta = \frac{Opposite}{Hypotenuse}$$
• $$\sec \theta = \frac{Hypotenuse}{Adjacent}$$
• $$\cos\theta = \frac{Adjacent}{Hypotenuse}$$
• $$\tan \theta =\frac{Opposite}{Adjacent}$$
• $$csc \theta = \frac{Hypotenuse}{Opposite}$$
• $$cot \theta = \frac{Adjacent}{Opposite}$$

The Reciprocal Identities are given as:

• $$cosec\theta =\frac{1}{\sin\theta }$$
• $$sec\theta =\frac{1}{\cos\theta }$$
• $$cot\theta =\frac{1}{\tan\theta }$$
• $$sin\theta =\frac{1}{csc\theta }$$
• $$cos\theta =\frac{1}{\sec\theta }$$
• $$tan\theta =\frac{1}{cot\theta }$$

Trigonometric Values of Special Angles

 Degree sin cos tan cot sec cosec 0∘ 0 1 0 Not Defined 1 Not Defined 30∘ $\frac{1}{2}$ $\frac{√3}{2}$ $\frac{1}{√3}$ √3 $\frac{2}{√3}$ 2 45∘ $\frac{1}{√2}$ $\frac{1}{√2}$ 1 1 √2 √2 60∘ $\frac{√3}{2}$ $\frac{1}{2}$ √3 $\frac{1}{√3}$ 2 $\frac{2}{√3}$ 90∘ 1 0 Not Defined 0 Not Defined 1

Product:

• $$\sin\: x\cdot \cos\:y=\frac{\sin(x+y)+\sin(x-y)}{2}$$
• $$\cos\: x\cdot \cos\:y=\frac{\cos(x+y)+\cos(x-y)}{2}$$
• $$\sin\: x\cdot \sin\:y=\frac{\cos(x+y)-\cos(x-y)}{2}$$

Sum to Product:

• $$\sin\: x+\sin\: y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$$
• $$\sin\: x-\sin\: y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$$
• $$\cos\: x+\cos\: y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$$
• $$\cos\: x-\cos\: y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$$<

Pythagorean

• Sin2x + Cos2x = 1
• 1 + tan2x = sec2x
• 1 + cot2x = cosec2x

• sinx = ∓√1cos2x
• tanx = ∓√sec2x-1
• cosx = ∓√1sin2x

Odd-Even

• Sin(-x)=-sin x
• cos(-x)=-cos x
• tan(-x)=-tan x
• cot(-x)=-cot x
• sec(-x)=-sec x
• cosec(-x)=-cosec x

Ratio or Quotient Identities are given as:

• $$Sinθ = Cosθ \times Tanθ$$
• $$Cosθ = Sinθ \times Cotθ$$
• $$Tanθ = \frac{Sinθ}{Cosθ}$$
• $$Cotθ = \frac{Cosθ}{Sinθ}$$

Periodicity

• $$sin(x+2\pi )=sin\; x$$
• $$cos(x+2\pi )=cos\; x$$
• $$tan(x+\pi )=tan\; x$$
• $$cot(x+\pi )=cot\; x$$

Co-function:

• $$sin(90^{\circ}-x)=cos\; x$$
• $$cos(90^{\circ}-x)=sin\; x$$
• $$tan(90^{\circ}-x)=cot\; x$$
• $$cot(90^{\circ}-x)=tan\; x$$

Sum/Difference:

• $$\sin (x + y) = \sin(x) \cos(y) + \cos(x) \sin(y)$$
• $$\cos(x + y) = \cos(x) \cos(y) – \sin(x) \sin(y)$$
• $$\tan(x+y)=\frac{\tan\: x+\tan\: y}{1-\tan\: x\cdot \tan\: y}$$
• $$\sin(x – y) = \sin(x) \cos(y) – \cos(x) \sin(y)$$
• $$\cos(x – y) = \cos(x) \cos(y) + \sin(x) \sin(y)$$
• $$\tan(x-y)=\frac{\tan\: x – \tan\: y}{1+\tan\: x\cdot tan\: y}$$

Double Angle:

• $$\sin(2x) = 2\sin(x).\cos(x)$$
• $$\cos(2x) = \cos^{2}(x) – \sin^{2}(x)$$
• $$\cos(2x) = 2 \cos^{2}(x) -1$$
• $$\cos(2x) = 1 – 2 \sin^{2}(x)$$
• $$\tan(2x) = \frac{[2\: \tan(x)]}{[1 -\tan^{2}(x)]}$$

Half Angle:

• $$\sin\frac{x}{2}=\pm \sqrt{\frac{1-\cos\: x}{2}}$$
• $$\cos\frac{x}{2}=\pm \sqrt{\frac{1+\cos\: x}{2}}$$
• $$\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}$$

#### Albegra Maths Formulas

• (a + b)2=a2 + 2ab + b2
• (a−b)2=a2−2ab + b2
• (a + b)(a – b)=a2 – b2
• (x + a)(x + b)=x2 + (a + b)x + ab
• (x + a)(x – b)=x2 + (a – b)x – ab
• (x – a)(x + b)=x2 + (b – a)x – ab
• (x – a)(x – b)=x2 – (a + b)x + ab
• (a + b)3=a3 + b3 + 3ab(a + b)
• (a – b)3=a3 – b3 – 3ab(a – b)
• (x + y + z)2=x2 + y2 + z2 + 2xy + 2yz + 2xz
• (x + y – z)2=x2 + y2 + z2 + 2xy – 2yz – 2xz
• (x – y + z)2=x2 + y2 + z2 – 2xy – 2yz + 2xz
• (x – y – z)2=x2 + y2 + z2 – 2xy + 2yz – 2xz
• x3 + y3 + z3 – 3xyz=(x + y + z)(x2 + y2 + z2 – xy – yz−xz)
• (x + a)(x + b)(x + c)=x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
• x3 + y3=(x + y)(x2 – xy + y2)
• x3 – y3=(x – y)(x2 + xy + y2)
• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4)
• (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4)
• a4 – b4 = (a – b)(a + b)(a2 + b2)
• a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
• If n is a natural number, an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
• If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
• If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
• (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….
• Laws of Exponents
(am)(an) = am+n
(ab)m = ambm
(am)n = amn
• Fractional Exponents
a0 = 1
aman=am−naman=am−n
amam = 1a−m1a−m
a−ma−m = 1am
 $$x^{2} + y^{2} = \frac{1}{2} \left [(x + y)^{2} + (x – y)^{2} \right ]$$ $$x^{2} + y^{2} + z^{2} -xy – yz – zx = \frac{1}{2} [(x-y)^{2} + (y-z)^{2} + (z-x)^{2}]$$ $$\mathbf{a_{1}x + b_{1}y + c_{1} = 0}$$ $$\mathbf{a_{2}x+ b_{2}y + c_{2}} = 0$$
 Distributive Property a ( b+c) = (a × b) + (a × c) Commutative Property of Addition a + b = b+a Commutative Property of Multiplication a× b = b×a Associative Property of Addition a + (b + c ) = ( a+ b ) +c Associative Property of Multiplication a ( b × c ) = ( a× b ) × c Additive Identity Property a +0 = a Multiplicative Identity Property a×1 = a Additive Inverse Property a+ ( -a) = 0 Multiplicative Inverse Property a × 1/a = 1 Zero Property of Multiplication a × 0 = 0
 If$$\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$$ then magnitude or length or norm or absolute value of $$\vec{a}$$ is $$\left | \overrightarrow{a} \right |=a=\sqrt{x^{2}+y^{2}+z^{2}}$$ A vector of unit magnitude is unit vector. If $$\vec{a}$$ is a vector then unit vector of $$\vec{a}$$ is denoted by $$\hat{a}$$ and $$\hat{a}=\frac{\hat{a}}{\left | \hat{a} \right |}$$ Therefore $$\hat{a}=\frac{\hat{a}}{\left | \hat{a} \right |}\hat{a}$$ Important unit vectors are $$\hat{i}, \hat{j}, \hat{k}$$, where $$\hat{i} = [1,0,0],\: \hat{j} = [0,1,0],\: \hat{k} = [0,0,1]$$ If $$l=\cos \alpha, m=\cos \beta, n=\cos\gamma,$$ then $$\alpha, \beta, \gamma,$$ are called directional angles of the vectors$$\overrightarrow{a}$$ and $$\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1$$ $$\vec{a}+\vec{b}=\vec{b}+\vec{a}$$ $$\vec{a}+\left ( \vec{b}+ \vec{c} \right )=\left ( \vec{a}+ \vec{b} \right )+\vec{c}$$ $$k\left ( \vec{a}+\vec{b} \right )=k\vec{a}+k\vec{b}$$ $$\vec{a}+\vec{0}=\vec{0}+\vec{a}$$, therefore $$\vec{0}$$ is the additive identity in vector addition. $$\vec{a}+\left ( -\vec{a} \right )=-\vec{a}+\vec{a}=\vec{0}$$, therefore $$\vec{a}$$ is the inverse in vector addition.

#### Probability and Set Theory Maths Formulas

• Commutative= $$A\cup B = B\cup A$$ and $$A\cap B = B\cap A$$
• Associative= $$A\cup (B\cup C) = A\cup (B\cup C)$$ and $$A\cap (B\cap C) = A\cap (B\cap C)$$
• Neutral element= $$A\cup \theta = A$$ and $$A\cap E = A$$
• Absorbing element= $$A\cup E = E$$ and $$A\cap \theta = \theta$$
• Distributive= $$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$$ and $$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$$
• De Morgan’s laws= $$\bar(A\cap B) = \bar A \cup \bar B$$ and $$\bar(A\cup B) = \bar A \cap \bar B$$
• Independent Events= $$P(A | B)=P(A)$$ and $$P(A\cap B)=P(A)×P(B)$$
• Conditional Probability= $$P(A | B)=\frac{P(A\cap B)}{P(B)}$$
• Laplace laws= $$P(A)=\frac{Number\;of\;ways\;it\;can\;happen}{Total\;Number\;of\;Outcomes}$$
• Complement of an Event= $$P (\bar A)=1 – P(A)$$
• Union of Events= $$P(A\cup B)=P(A)+P(B)−P(A\cap B)$$

#### Statistics Maths Formulas

 $$Direct\; Method: x̅ = \frac{\sum_{i=1}^{n}f_i x_i}{\sum_{i=1}^{n}f_i}$$ where fi xi is the sum of observations from value i = 1 to n And fi is the number of observations from value i = 1 to n $$Assumed\; mean\; method : x̅ = a+\frac{\sum_{i=1}^{n}f_i d_i}{\sum_{i=1}^{n}f_i}$$ $$Step \;deviation \;method : x̅ = a+\frac{\sum_{i=1}^{n}f_i u_i}{\sum_{i=1}^{n}f_i}\times h$$ $$Mode = l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2} \times h$$ $$Range = Largest\; Value – Smallest\; Value$$ $$Median = l+\frac{\frac{n}{2} – cf}{f} \times h$$ $$Variance = \sigma ^{2} = \frac{\sum (x- \bar{x})^{2}}{n}$$Where, x = Items given, x¯ = Mean, n = Total number of items $$Standard\;Deviation \; \sigma = \sqrt{\frac{\sum (x-\bar{x})^{2}}{n}}$$ Where, x = Items given, x¯ = Mean, n = Total number of items

#### Exponents and Powers Maths Formulas

• Power zero = $$a^{ 0 } = 1$$
• Power one = $$a^{ 1 } = a$$
• Fraction formula = $$\sqrt{ a } = a^{ \frac{ 1 }{ 2 }}$$
• Reverse formula = $$\sqrt{ n } { a } = a^{\frac{ 1 } { n }}$$
• Negative power value = $$a^{ -n } = \frac{ 1 }{ a^{n} }$$
• Fraction formula = $$a^{n} = \frac{1}{ a^{ -n } }$$
• Product formula = $$a^{m}a^{n} = a^{ m + n }$$
• Division Formula = $$\frac{ a^{ m }}{ a^{ n }} = a ^{ m-n }$$
• Power of Power formula = $$(a^{ m })^{ p } = a^{ mp }$$
• Power distribution Formula = $$(a^ { m }c^{ n })^{ x } = a ^ { mx } c ^{ nx }$$
• The Power distribution Formula = $$\left ( \frac {a ^{ m }}{c^{ n }} \right )^{x} = \frac{a^{ mx }}{c^{ nx }}$$
• Power Even Number = $$(-1)^{Even Number} = 1$$
• Power Odd Number = $$(-1)^{Odd Number} = -1$$
• Product of Power Formula = $$(ab)^m = a^m \times b^m$$

#### Complex Number Maths Formulas

• Complex number = $$z=a+bi$$
• Conjugate = $$\bar z =a−bi$$
• Symmetry = $$−z=−a−bi$$
• Equality = a+bi=c+di⇔a=c and b=d
• Addition = $$(a+bi)+(c+di)=(a+c)+(b+d)i$$
• Subtraction = $$(a+bi)−(c+di)=(a−c)+(b−d)i$$
• Multiplication = $$(a+bi)\times(c+di)=(ac−bd)+(ad+bc)i$$
• Division = $$\frac{(a+bi)}{(c+di)} = \frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{ac+bd}{c^{2}+d^{2}} + \frac{bc-ad}{c^{2}+d^{2}}\times i$$
• Multiplication Conjugates = $$(a+bi)(a+bi)=a^{2}+b^{2}$$
Powers of Complex Numbers
• in = i, if n = 4a+1, i.e. one more than the multiple of 4.

Example – i1=i; i5=i; i9=i; i4a+1;

• in = -1, if n = 4a+2, i.e. two more than the multiple of 4.

Example – i2= -1; i6= -1; i10= -1; i4a+2;

• in = -i, if n = 4a+3, i.e. Three more than the multiple of 4.

Example – i3= -i; i7= -i; i11= -i; i4a+3;

• in = -1, if n = 4a, i.e. the multiple of 4.

Example – i4= 1; i8= 1; i12= 1; i4a;

#### Logic Maths Formulas

Conjunction      Disjunction      Implication
pqp∧ q   pqp∨q   pqp ⇒ q
VVV   VVV   VVV
VFF   VFV   VFF
FVF   FVV   FVV
FFF   FFF   FFV
 Law of noncontradiction p∧~p⇔F Law of the excluded middle p∨~p⇔V Double Negation ~(~p)⇔p Commutativity Conjunction p∧q⇔q∧p Disjunction p∨q⇔q∨p Associativity Conjunction (p∧q)∧r⇔p∧(q∧r) Disjunction (p∨q)∨r⇔p∨(q∨r) Neutral Element Conjunction p∧V⇔p Disjunction p∨F⇔p Absorbing Element Conjunction p∧F⇔F Disjunction p∨V⇔V Idempotence Conjunction p∧p⇔p Disjunction p∨p⇔p Distributive Property Conjunction over Disjunction p∧(q∨r)⇔(p∧q)∨(p∧r) Disjunction over Conjunction p∨(q∧r)⇔(p∨q)∧(p∨r) Properties of Implication Transitive (p⇒q)∧(q⇒r)⇒(p⇒r) Implication and Disjunction (p⇒q)⇔~p∨q Negation ~(p⇒q)⇔p∧~q Contrapositive of an Implication (p⇒q)⇔(~q⇒~p) Properties of Equivalence Double implication (p⇔q)⇔[(p⇒q)∧(q⇒p)] Transitive [(p⇔q)∧(q⇔r)]⇒(p⇔r) Negation ~(p⇔q)⇔[(p∧~q)∨(q∧~p)] De Morgan’s laws Negation of a Conjunction ~(p∧q)⇔~p∨~q Negation of a Disjunction ~(p∨q)⇔~p∧~q De Morgan’s laws Negation of Universal Quantifier ~(∀x,p(x))⇔∃x:~p(x) Negation of Existential Quantifier ~(∃x:p(x))⇔∀x,~p(x)

#### Limits Maths Formulas

1) If $$\lim_{x \to a} f(x) = l\; and \; \lim_{x \to a} g(x) = m$$, then

• $$\lim_{x \to a} \left[ {f(x) \pm g(x)} \right] = l \pm m$$
• $$\lim_{x \to a} f(x) \cdot g(x) = l \cdot m$$
• $$\lim_{x \to a} \frac{{f(x)}}{{g(x)}} = \frac{l}{m}$$, where \)m \ne 0\)
• $$\lim_{x \to a} c{\text{ }}f(x) = c{\text{ }}l$$
• $$\lim_{x \to a} \frac{1}{{f(x)}} = \frac{1}{l}$$, where \)l \ne 0\)

2) $$\lim_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^n} = e$$, where \)n\) is a real number.

3) $$\lim_{n \to 0} {\left( {1 + n} \right)^{\frac{1}{n}}} = e$$, where \)n\) is a real number.

4) $$\lim_{x \to 0} \frac{{Sinx}}{x} = 1$$, where \)x\) is measured in radians.

5) $$\lim_{x \to 0} \frac{{Tanx}}{x} = 1$$

6) $$\lim_{x \to 0} \frac{{Cosx – 1}}{x} = 0$$

7) $$\lim_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}} = n{a^{n – 1}}$$

8) $$\lim_{x \to 0} \frac{{{a^n} – 1}}{x} = \ln a$$

#### Logarithms Maths Formulas

Properties

• $$\log_{b}1 = 0$$        (∵ b^0 = 1)
• $$\log_{b}b = 1$$        (∵ b1 = b)
• $$y = \ln x \quad \Rightarrow e^y = x$$
• $$x = e^y \quad \Rightarrow \ln x = y$$
• $$x = \ln e^x = e^{\ln x}$$
• $$b^{\log_{b}x} = x$$
• $$\log_{b}{b^y} = y$$

Laws of Logarithms

1. $$\log_{b}{M \times N} = \log_b{{M}} + \log_b{{N}}$$ (where b, M, N are positive real numbers and b ≠ 1)

2. $$\log_{b}\frac{M}{N} = \log_b{{M}} – \log_b{{N}}$$(where b, M, N are positive real numbers and b ≠ 1)

3. $$\log_{b}M^c = c \ \log_b{{M}}$$ (where b and M are positive real numbers , b ≠ 1, c is any real number)

4. $$\log_{b}M = \frac{\log {M}}{\log b} = \frac{\ln {M}}{\ln b} = \frac{\log_{k} {M}}{\log_{k} b}$$ (where b, k and M are positive real numbers, b ≠ 1, k ≠ 1)

5. $$\log_{b}a = \frac{1}{\log_{a}b}$$ (where a and b are positive real numbers, a ≠ 1, b ≠ 1)

6. If $$\log_{b}{M}=\log_{b}{N}$$, then M = N (where b, M and N are positive real numbers and b ≠ 1).

#### Related Math Formulas

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