Mathematics Formulae

Mathematics Formulae

for School Students

M. D. Raghu

FORMULAEFOR STUDENTSMATHEMATICS

©2014 MDR www.learningforknowledge.com/glg

𝒙𝒎 .𝒙𝒏=𝒙𝒎+𝒏

𝒙𝒎

𝒙𝒏 =𝒙𝒎−𝒏

(𝒙𝒎)𝒏=𝒙𝒎𝒏

𝒙𝒑𝒒=

𝒒√𝒙𝒑

Algebra

Indices Logarithms

log𝒃𝒙𝒚=log𝒃𝒙+ log𝒃𝒚

log𝒃𝒙𝒚

=log𝑏 𝒙− log𝒃 𝒚

log𝒃𝒙𝒏=𝒏 log𝒃𝒙

log𝒂𝒙=log𝑏𝒙log𝒃𝒂



Algebra

Quadratic Equations

(𝑎2−𝑏2 )=(𝑎−𝑏)(𝑎+𝑏)

(𝑎−𝑏)2=𝑎2−2𝑎𝑏+𝑏2(𝑎+𝑏 )2=𝑎2+2𝑎𝑏+𝑏2

Binomial Expansion



Algebra

Binomial Theorem

h𝑤 𝑒𝑟𝑒𝑛𝐶𝑟=𝑛 !

(𝑛−𝑟 )!×𝑟 !



Arithmetic

𝒕𝒏=𝒂𝒓𝒏−𝟏

𝒓=𝒕𝒏𝒕𝒏−𝟏

Arithmetic Series

𝑎 ,𝑎+𝑑 ,𝑎+2𝑑 ,…Geometric Series

𝑎 ,𝑎𝑟 ,𝑎𝑟2 ,…



Arithmetic

Complex Numbers

(𝒛𝟏± 𝒛𝟐 )=𝒛𝟏 . 𝒛𝟐

𝒛 𝒛=|𝒛|𝟐

|𝒛𝟏 . 𝒛𝟐|=|𝒛𝟏||𝒛𝟐|



Arithmetic

Parametric Complex Numbers

De Moivre’s Theorem

(𝐜𝐨𝐬𝜽+𝒊𝐬𝐢𝐧 𝜽 )𝒏=𝐜𝐨𝐬𝒏𝜽+𝒊𝐬𝐢𝐧𝒏𝜽

𝒛=𝒓 (𝐜𝐨𝐬𝜽+𝒊𝐬𝐢𝐧 𝜽 )

𝒓=|𝒛|=√𝒙𝟐+𝒚𝟐



Analytical Geomtery

Line Coordinates and Gradients



Analytical Geomtery

Line Equations



𝒔𝒊𝒏𝜽=𝒚𝒓

𝒄𝒐𝒔 𝜽=𝒙𝒓

𝒕𝒂𝒏𝜽=𝒚𝒙

𝒄𝒐𝒔𝒆𝒄 𝜽=𝒓𝒚

𝒔𝒆𝒄 𝜽=𝒓𝒙

𝒄𝒐𝒕 𝜽=𝒙𝒚

Trigonometry

Ratios

x

y

r

θ



cos−1𝒙𝒓

=𝜽

sin−1𝒚𝒓

=𝜽

tan−1𝒚𝒙

=𝜽 cot−1𝒙𝒚

=𝜽

c o se c−1𝒓𝒚

=𝜽

sec−1𝒓𝒙

=𝜽

Trigonometry

Inverse of Ratios

x

y

r

θ



tan𝜽=¿sin𝜽cos𝜽

¿

tan 2𝜽+1=sec 2𝜽

sin 2𝜽+cos2𝜽=1

cosec 𝜽=𝟏sin 𝜽

Trigonometry

Identities

sec𝜽=𝟏cos𝜽

cot 𝜽=cos𝜽sin 𝜽

cot 2𝜽+1=cosec 2𝜽

tan 𝜽×cot 𝜽=1



Trigonometry

Products

2cos 𝐴 cos𝐵=cos (𝐴−𝐵 )+cos ( 𝐴+𝐵 )

2cos 𝐴 sin𝐵=sin ( 𝐴+𝐵 )− sin ( 𝐴−𝐵 )

2sin 𝐴 cos𝐵=sin ( 𝐴+𝐵 )+sin ( 𝐴−𝐵 )

2sin 𝐴sin 𝐵=cos ( 𝐴−𝐵 )− cos ( 𝐴+𝐵 )



Trigonometry

Sums

cos𝐶+cos𝐷=2cos(𝐶+𝐷 )2

× cos(𝐶−𝐷 )2

sin𝐶−sin𝐷=2cos(𝐶+𝐷 )2

×sin(𝐶−𝐷 )2

cos𝐶− cos𝐷=−2sin(𝐶+𝐷 )2

×sin(𝐶−𝐷 )2

sin𝐶+sin𝐷=2sin(𝐶+𝐷 )2

×cos(𝐶−𝐷 )2



tan ( 𝐴−𝐵 )= tan 𝐴− tan𝐵1+ tan 𝐴 tan𝐵

sin ( 𝐴−𝐵 )=sin 𝐴cos 𝐵− cos 𝐴 sin𝐵

cos ( 𝐴+𝐵 )=cos 𝐴cos 𝐵− sin 𝐴sin𝐵

sin ( 𝐴+𝐵 )=sin 𝐴 cos𝐵+cos 𝐴sin𝐵

cos ( 𝐴−𝐵 )=cos 𝐴 cos𝐵+sin 𝐴sin𝐵

tan ( 𝐴+𝐵 )= tan 𝐴+ tan𝐵1− tan 𝐴 tan𝐵

Trigonometry

Compound Angles



𝑨=𝟏𝟐𝒓𝟐𝜽

𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔=𝟏𝟖𝟎°

𝒂𝟐=𝒃𝟐+𝒄𝟐−𝟐𝒃𝒄cos𝑨

𝒂sin 𝑨

=𝒃

sin𝑩=

𝒄sin𝑪

𝒔=𝒓 𝜽

𝑨=𝟏𝟐𝒂𝒃sin𝑪

Trigonometry

Rules

Sine Rule

Cosine Rule

Area of a triangle

Radians

Length of arc

Area of sector



Trigonometry

General Solutions

sin (90−𝜃)=cos𝜃

cos (90−𝜃 )=sin𝜃

tan (90−𝜃 )=cot 𝜃

Complementary angles Multiple angles



𝑃=2𝑙+2𝑏 𝐴=𝑙𝑏

𝐶=2𝜋 𝑟 𝐴=𝜋 𝑟2

Geometry

Perimeter AreaShape

a

b

ch

l

b

r

𝑃=𝑎+𝑏+𝑐 𝐴=12𝑏 h

𝑠=12(𝑎+𝑏+𝑐) 𝐴=√𝑠 (𝑠−𝑎 ) (𝑠−𝑏) (𝑠−𝑐 )



Geometry

Surface AreaShape Volume

𝑉=43𝜋 𝑟3

𝑉=𝜋 𝑟2h

Sphere

Cylinder

Cone

𝑉=13𝜋𝑟2h



Geometry

Surface AreaShape Volume

𝑉=𝑙3

𝑉= h𝑙𝑏

𝑉= (𝐵𝐴 )h

Cube

Cuboid

Prism



Geometry

Conic Sections

𝑥2

𝑎2+ 𝑦2

𝑏2=1

AlgebraicShape Parametric

Circle

Ellipse

Hyperbolae

𝑥2

𝑎2−𝑦 2

𝑏2=1 𝑥=𝑎 sec 𝜃 , 𝑦=𝑏 tan 𝜃



Geometry

Conic Sections

𝑥2=4𝑎𝑦

Algebraic EquationShape

Parabola

General Equation to a Conic



Calculus

Limits

lim𝑥→ 0

sin𝜃𝜃

=1 lim𝑛→∞

𝑎𝑥𝑛

=𝑎

Incremental Limits

𝑑𝑦𝑑𝑥

= 𝑓 ′ (𝑥 )=limh→0

𝑓 (𝑥+h )− 𝑓 (𝑥)h



Calculus

Differentiation

𝐺𝑖𝑣𝑒𝑛 𝑦= 𝑓 (𝑥 )𝑎𝑛𝑑 𝑦 ′= 𝑓 ′ (𝑥 )=𝑑𝑦𝑑𝑥

c 0



Calculus

Differentiation Rules

𝐺𝑖𝑣𝑒𝑛𝑢= 𝑓 (𝑥 )𝑎𝑛𝑑 𝑣=𝑔 (𝑥 )

Product Rule Quotient Rule



Calculus

Differentiation Rules

𝐺𝑖𝑣𝑒𝑛𝑢=𝑔 (𝑥 )𝑎𝑛𝑑 𝑦= 𝑓 (𝑢)

𝑑𝑦𝑑𝑥

=𝑑𝑦𝑑𝑢.𝑑𝑢𝑑𝑥

Chain Rule or Composite Function Rule

Gradient

𝑑𝑦𝑑𝑥

𝑜𝑓 𝑓 (𝑥 )𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 (𝑥1 , 𝑦1 )=𝑚



Calculus

Integration

k



Calculus

Integration Rules

∫𝑎

𝑏

𝑓 ′ (𝑥 )ⅆ 𝑥= 𝑓 (𝑏)− 𝑓 (𝑎)

∫ 𝑓 (𝑥 )𝑔 (𝑥 )ⅆ 𝑥= 𝑓 (𝑥 )∫𝑔 (𝑥 )ⅆ 𝑥−∬𝑔 (𝑥 )ⅆ 𝑥𝑑𝑓 (𝑥)ⅆ 𝑥

Volume Integral

𝑉=𝜋∫𝑎

𝑏

𝑦2ⅆ 𝑥



Author: M. D. Raghu

Email: [email protected]

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Mathematics Formulae