Math Results And Formulas

Factors and Products Formulas

$${(a + b)^2} = {a^2} + 2ab + {b^2}$$ $${(a – b)^2} = {a^2} – 2ab + {b^2}$$ $${(a +… Click here to read more

Ratio and Proportion Formulas

1. If $$\frac{a}{b} = \frac{c}{d}$$ then $$ad = bc$$ 2. If $$\frac{a}{b} = \frac{c}{d}$$ then $$\frac{a}{c} = \frac{b}{d}$$ 3. If… Click here to read more

Exponents Formulas

1. If $$p$$ is a positive integer and $$a \in \mathbb{R}$$, then $${a^p} = a \cdot a \cdot a \cdots… Click here to read more

Logarithm Formulas

1. $$y = {\log _a}x$$ if and only if $$x = {a^y}$$, $$x > 0$$ and $$y \in \mathbb{R}$$, $$a$$… Click here to read more

Important Math Series

1. $${e^x} = 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + \cdots $$ 2. $$\ln (1 + x) = x… Click here to read more

Math Series Results

1. $$1 + 2 + 3 +\cdots + n = \frac{{n(n + 1)}}{2}$$ 2. $${1^2} + {2^2} + {3^2} +… Click here to read more

Results and Formulas of Equations

If $$\alpha $$ and $$\beta $$ are the roots of the Quadratic Equation $$a{x^2} + bx + c = 0$$,… Click here to read more

Formulas of Sequence and Series

The $$nth$$ term $${a_n}$$ of the Arithmetic Progression (A.P) $$a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots $$ is given… Click here to read more

Results of Absolute Value

Consider that $$a,b \in \mathbb{R}$$, then 1. $$\left| a \right| \geqslant 0$$ and $$\left| a \right| = 0 \Leftrightarrow a… Click here to read more

Formulas and Results of Complex Numbers

$$z = (a,b) = a + ib,{\text{ }}i = (0,1)$$ $$i = \sqrt { – 1} ,{\text{ }}{i^2} =- 1,{\text{}}{i^3}… Click here to read more

Formulas of Useful Limits

1) If $$\mathop {\lim }\limits_{x \to a} f(x) = l$$ and $$\mathop {\lim }\limits_{x \to a} g(x) = m$$, then… Click here to read more

Basic Derivative Results

If $$y = f(x)$$, then 1) $$\frac{{dy}}{{dx}} = f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) – f(x)}}{h}$$… Click here to read more

Basic Formulas of Derivatives

General Derivative Formulas: 1) $$\frac{d}{{dx}}(c) = 0$$ where $$c$$ is any constant. 2) $$\frac{d}{{dx}}{x^n} = n{x^{n – 1}}$$ is called… Click here to read more

Results of Higher Derivatives

1) \[{y_n} = \frac{{{d^n}}}{{d{x^n}}}{(ax + b)^m} = \frac{{m!}}{{(m – n)!}}{a^n}{(ax + b)^{m – n}}\] 2) \[{y_n} = \frac{{{d^n}}}{{d{x^n}}}\frac{1}{{(ax + b)}}… Click here to read more

Formulas of Integration

1) \[\int {1dx = x + c} \] 2) \[\int {adx = ax + c} \] where $$a$$ is any… Click here to read more

Reduction Formulas of Integration

1) \[\int {{{\sin }^n}xdx = – \frac{{\cos x{{\sin }^{n – 1}}x}}{n} + \frac{{n – 1}}{n}\int {{{\sin }^{n – 2}}xdx} }… Click here to read more

Results and Formulas of Definite Integrals

1) $$\int\limits_a^b {F'(x)dx = F(a) – F(b)} $$ is called the Fundamental Theorem of Integral Calculus. 2) \[\int\limits_a^b {f(x)dx} = –… Click here to read more

Results and Formulas of Beta and Gamma Integrals

1) \[\beta (m.n) = \int\limits_0^1 {{x^{m – 1}}{{(1 – x)}^{n – 1}}dx} \] is called the Beta Integral. 2) \[\Gamma… Click here to read more

Set Theory Formula

Consider that $$A$$,$$B$$ and $$C$$ are the sets, then 1. \[A \cup A = A\] 2. \[A \cap A =… Click here to read more

Set Theory Results

Consider that $$A$$,$$B$$ and $$C$$ are the sets, then 1. $$A \subseteq A \cup B$$ and $$B \subseteq A \cup… Click here to read more

Math Symbols

MATH SYMBOLS Symbols Description Symbols Description $$ = $$ is equal to $$\vartriangle $$ triangle $$ \ne $$ is not… Click here to read more

The Greek Alphabet

The Greek Alphabet Symbols Description Symbols Description $${\rm A}$$ $$\alpha $$ Alpha $$\Xi $$ $$\xi $$ XI $${\rm B}$$ $$\beta… Click here to read more

Mathematical Constants

$$\pi = 3.14159$$ $${\pi ^2} = 9.8696$$ $$\sqrt \pi = 1.7724$$ $$\frac{1}{\pi } = 0.3183$$ $$e = 2.7182$$ $${\log _{10}}\pi… Click here to read more

Mathematical Facts

The weight of one cu. ft water = $$62.425{\text{ }}lb$$ The velocity of light in a vacuum $$ = 2.99776… Click here to read more

Weights and Measures

Length: 10 millimeters = 1 centimeter 10 centimeters = 1 decimeter 10 decimeters = 1 meter 10 meters = 1… Click here to read more

Some Basic Types and Results of Triangles

1. An angle whose measurement is of $${90^ \circ }$$ is called a RIGHT ANGLE. 2. An angle whose measurement is greater than… Click here to read more

Results of Angles with Degree and Radian

1) ANGLE: The union of two non colinear rays which have a common end point is called an angle. 2)… Click here to read more

Fundamental Trigonometric Ratios

1) $$Sin\theta = \frac{{perpendicular}}{{hypotenuse}} = \frac{y}{r}$$ 2) $$Cos\theta = \frac{{base}}{{hypotenuse}} = \frac{x}{r}$$ 3) $$Tan\theta = \frac{{perpendicular}}{{base}} = \frac{y}{x} = \frac{{Sin\theta… Click here to read more

Results of Trigonometric Ratios of Allied Angles

$$\alpha $$ $$Sin\alpha $$ $$Cos\alpha $$ $$Tan\alpha $$ $$Cot\alpha $$ $$Sec\alpha $$ $$Co\sec \alpha $$ $$ – \theta $$ $$… Click here to read more

Fundamental General Identities Involving Trigonometric Ratios

\[Si{n^2}\theta + Co{s^2}\theta = 1\] \[1 + Ta{n^2}\theta = Se{c^2}\theta \] \[1 + Co{t^2}\theta = Cose{c^2}\theta \] \[Sin(\alpha + \beta… Click here to read more

Formulas of Sum and Product of Trigonometric Ratios

\[Sina + Sinb = 2Sin\left( {\frac{{a + b}}{2}} \right)Cos\left( {\frac{{a – b}}{2}} \right)\] \[Sina – Sinb = 2Cos\left( {\frac{{a +… Click here to read more

Results and Formulas of Plane Triangles

Consider the triangle $$\Delta ABC$$, having the angles $$\alpha ,\beta ,\gamma $$ and sides $$a,b,c$$ as shown in the figure…. Click here to read more

Circles Connected to a Triangle

1) If $$r$$ denotes in-radius, then \[r = \sqrt {\frac{{(s – a)(s – b)(s – c)}}{s}} = \frac{\Delta }{s}\] \[r… Click here to read more

Formulas for Sum and Difference of Inverse Trigonometric Functions

\[Si{n^{ – 1}}a + Si{n^{ – 1}}b = Si{n^{ – 1}}(a\sqrt {1 – {b^2}} + b\sqrt {1 – {a^2}} )\]… Click here to read more

Formulas and Results of Hyperbolic Functions

\[Sinhx = \frac{{{e^x} – {e^{ – x}}}}{2}\] \[Coshx = \frac{{{e^x} + {e^{ – x}}}}{2}\] \[Tanhx = \frac{{{e^x} – {e^{ –… Click here to read more

Formulas of Inverse Hyperbolic Functions

\[Sin{h^{ – 1}}x = {\log _e}\left| {x + \sqrt {{x^2} + 1} } \right|\] \[Cos{h^{ – 1}}x = {\log _e}\left|… Click here to read more

Change of Hyperbolic Functions

1) \[Sinh{\text{ }}ix = iSinx\] 2) \[Cosh{\text{ }}ix = Cosx\] 3) \[Tanh{\text{ }}ix = iTanx\] 4) \[iSinhx = Sin{\text{ }}ix\]… Click here to read more

Formulas of Trigonometric and Logarithmic Functions

1) \[Sinx = \frac{{{e^{ix}} – {e^{ – ix}}}}{{2i}}\] 2) \[Cosx = \frac{{{e^{ix}} + {e^{ – ix}}}}{2}\] 3) \[Tanx = \frac{{{e^{ix}}… Click here to read more

Formulas of Inverse Trigonometric Functions

1) \[Si{n^{ – 1}}x = \frac{1}{i}Log(ix + \sqrt {1 – {x^2}} )\] 2) \[Co{s^{ – 1}}x = \frac{1}{i}Log(x + \sqrt… Click here to read more

Formulas and Results of Straight Lines

Consider two points $$P\left( {{x_1},{y_1}} \right)$$ and $$Q\left( {{x_2},{y_2}} \right)$$, then: 01. The distance formula \[\left| {PQ} \right| = \sqrt… Click here to read more

Results and Formulas of a Circle

01. The equation of a circle having the center at $$O\left( {0,0} \right)$$ and radius $$r$$ is \[{x^2} + {y^2}… Click here to read more

Tangent and Normal Formulas

The formulas of tangent and normal to any curve at a given point are listed below. $${\left. {\frac{{dy}}{{dx}}} \right|_p}$$ is… Click here to read more

Equations of Tangent and Normal to the Circle

Here we list the equations of tangent and normal for different forms of a circle and also list the condition… Click here to read more

Equations of Tangent and Normal to the Parabola

Here we list the equations of tangent and normal for different forms of a parabola. The euation of tangent to… Click here to read more

Equations of Tangent and Normal to the Ellipse

Here we list the equations of tangent and normal for different forms of ellipses. We also define parallel chords and… Click here to read more

Equations of Tangent and Normal to the Hyperbola

Here we list the equations of tangent and normal for different forms of a hyperbola. The equation of tangent to… Click here to read more

Polar Equation of a Conic

Some important results and formulas regarding the polar equation of a conic are listed here. 1. The polar equation of a… Click here to read more

Formulas of Arc Length

Some important results and formulas regarding the arc length of the curve are listed here. 1. The arc length for… Click here to read more

Equations of Asymptotes

The results and formulas related to asymptotes are listed below. Asymptotes a. Obtain $${\phi _n}\left( m \right)$$ by putting $$x… Click here to read more

Formulas of Curvature and Radius of Curvature

The commonly used results and formulas of curvature and radius of curvature are as shown below: 1. Curvature $${\rm K}$$… Click here to read more

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