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Mathematics Formulae
- 1. Mathematics Formulae for School Students M. D. Raghu
- 2. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝒙 𝒎. 𝒙 𝒏 = 𝒙 𝒎+𝒏 𝒙 𝒎 𝒙 𝒏 = 𝒙 𝒎−𝒏 𝒙 𝒎 𝒏 = 𝒙 𝒎𝒏 𝒙 𝒑 𝒒 = 𝒒 𝒙 𝒑 Algebra Indices Logarithms 𝑰𝒇 𝒚 = 𝒃 𝒙 𝒕𝒉𝒆𝒏 𝒙 = log 𝒃 𝒚 log 𝒃 𝒙𝒚 = log 𝒃 𝒙 + log 𝒃 𝒚 log 𝒃 𝒙 𝒚 = log 𝑏 𝒙 − log 𝒃 𝒚 log 𝒃 𝒙 𝒏 = 𝒏 log 𝒃 𝒙 log 𝒂 𝒙 = log 𝑏 𝒙 log 𝒃 𝒂
- 3. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝐼𝑓 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑡ℎ𝑒𝑛 𝑥 = −𝑏 ± 𝑏2 − 4𝑎𝑐 2𝑎 Algebra Quadratic Equations 𝑎2 − 𝑏2 = (𝑎 − 𝑏)(𝑎 + 𝑏) 𝑎 − 𝑏 2 = 𝑎2 − 2𝑎𝑏 + 𝑏2 𝑎 + 𝑏 2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 Binomial Expansion
- 4. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝑎 + 𝑏 𝑛 = 𝑛 𝐶0 𝑎 𝑛 + 𝑛 𝐶1 𝑎 𝑛−1 𝑏 + ⋯ + 𝑛 𝐶 𝑟 𝑎 𝑛−𝑟 𝑏 𝑟 + ⋯ + 𝑛 𝐶 𝑛 𝑏 𝑛 Algebra Binomial Theorem 𝑤ℎ𝑒𝑟𝑒 𝑛 𝐶 𝑟 = 𝑛! 𝑛 − 𝑟 ! × 𝑟! 1 + 𝑥 𝑛 = 1 + 𝑛 𝐶1 𝑥 + ⋯ + 𝑛 𝐶 𝑟 𝑥 𝑟 + ⋯ 𝐼𝑓 𝑥 < 1, 𝑡ℎ𝑒𝑛 𝑥 𝑟 → 0 𝑎𝑠 𝑟 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠, 𝑟 ∈ 𝑁
- 5. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Arithmetic 𝒏 𝒕𝒉 𝒕𝒆𝒓𝒎 𝒕 𝒏 = 𝒂 + 𝒏 − 𝟏 𝒅 𝑴𝒆𝒂𝒏 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒅 = 𝒕 𝒏+𝟏 − 𝒕 𝒏 𝑺𝒖𝒎 𝒐𝒇 𝒏 𝒕𝒆𝒓𝒎𝒔 𝒔 𝒏 = 𝒏 𝟐 𝟐𝒂 + 𝒏 − 𝟏 𝒅 𝒔 𝒏 = 𝒂 𝟏−𝒓 𝒏 𝟏−𝒓 , 𝒓 ≠ 𝟏 𝒕 𝒏 = 𝒂 𝒓 𝒏−𝟏 𝒓 = 𝒕 𝒏 𝒕 𝒏−𝟏 𝒔∞ = 𝒂 𝟏−𝒓 , 𝒓 < 𝟏 Arithmetic Series 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, … Geometric Series 𝑎, 𝑎𝑟, 𝑎𝑟2, …
- 6. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Arithmetic 𝒛 = 𝒙 − 𝒊𝒚 𝒘𝒉𝒆𝒓𝒆 𝒛 𝒊𝒔 𝒄𝒐𝒏𝒋𝒖𝒈𝒂𝒕𝒆 𝒐𝒇 𝒛 𝑹𝒆𝒂𝒍 𝒑𝒂𝒓𝒕 𝒛 = 𝒛 + 𝒛 𝟐 𝑰𝒎𝒂𝒈𝒊𝒏𝒂𝒓𝒚 𝒑𝒂𝒓𝒕 𝒛 = 𝒛 − 𝒛 𝟐𝒊 𝒛 = 𝒙 + 𝒊𝒚 𝒘𝒉𝒆𝒓𝒆 𝒊 = −𝟏 Complex Numbers 𝒛 𝟏 ± 𝒛 𝟐 = 𝒛 𝟏 . 𝒛 𝟐 𝒛 𝒛 = 𝒛 𝟐 𝒛 𝟏. 𝒛 𝟐 = 𝒛 𝟏 𝒛 𝟐
- 7. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Arithmetic Parametric Complex Numbers De Moivre’s Theorem 𝒓 𝐜𝐢𝐬 𝜽 𝒏 = 𝒓 𝒏 𝐜𝐢𝐬 𝒏𝜽 𝒘𝒉𝒆𝒓𝒆 𝐜𝐢𝐬 𝜽 = 𝐜𝐨𝐬 𝜽 + 𝒊 𝐬𝐢𝐧 𝜽 , 𝒏 ∈ 𝑰 𝐜𝐨𝐬 𝜽 + 𝒊 𝐬𝐢𝐧 𝜽 𝒏 = 𝐜𝐨𝐬 𝒏𝜽 + 𝒊 𝐬𝐢𝐧 𝒏𝜽 𝒛 = 𝒓 𝐜𝐨𝐬 𝜽 + 𝒊 𝐬𝐢𝐧 𝜽 𝒂𝒓𝒈. 𝒛 = 𝜽 𝒘𝒉𝒆𝒓𝒆 cos 𝜽 = 𝒙 𝒓 𝒂𝒏𝒅 sin 𝜽 = 𝒚 𝒓 𝒓 = 𝒛 = 𝒙 𝟐 + 𝒚 𝟐
- 8. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝑴𝒊𝒅𝒑𝒐𝒊𝒏𝒕 𝒙 𝟏 + 𝒙 𝟐 𝟐 , 𝒚 𝟏 + 𝒚 𝟐 𝟐 𝑮𝒓𝒂𝒅𝒊𝒆𝒏𝒕 𝒎 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒙 𝟏 − 𝒙 𝟐 𝟐 + 𝒚 𝟏 − 𝒚 𝟐 𝟐 𝑷𝒂𝒓𝒂𝒍𝒍𝒆𝒍 𝑳𝒊𝒏𝒆𝒔 𝒎 𝟏 = 𝒎 𝟐 𝑷𝒆𝒓𝒑𝒆𝒏𝒅𝒊𝒄𝒖𝒍𝒂𝒓 𝑳𝒊𝒏𝒆𝒔 𝒎 𝟏 𝒎 𝟐 = −𝟏 Analytical Geomtery Line Coordinates and Gradients
- 9. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝑨𝒙𝒆𝒔 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒙 𝒂 + 𝒚 𝒃 = 𝟏 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒐𝒇 𝒑𝒐𝒊𝒏𝒕 𝒙 𝟏, 𝒚 𝟏 𝒇𝒓𝒐𝒎 𝒍𝒊𝒏𝒆 𝒂𝒙 + 𝒃𝒚 + 𝒄 = 𝟎 𝒂𝒙 𝟏 + 𝒃𝒚 𝟏 + 𝒄 𝒂 𝟐 + 𝒃 𝟐 𝑺𝒊𝒏𝒈𝒍𝒆 𝑷𝒐𝒊𝒏𝒕 𝒚 − 𝒚 𝟏 = 𝒎 𝒙 − 𝒙 𝟏 𝑺𝒍𝒐𝒑𝒆 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒚 = 𝒎𝒙 + 𝒄 Analytical Geomtery Line Equations
- 10. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝒔𝒊𝒏 𝜽 = 𝒚 𝒓 𝒄𝒐𝒔 𝜽 = 𝒙 𝒓 𝒕𝒂𝒏 𝜽 = 𝒚 𝒙 𝒄𝒐𝒔𝒆𝒄 𝜽 = 𝒓 𝒚 𝒔𝒆𝒄 𝜽 = 𝒓 𝒙 𝒄𝒐𝒕 𝜽 = 𝒙 𝒚 Trigonometry Ratios x y r θ
- 11. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg cos−1 𝒙 𝒓 = 𝜽 sin−1 𝒚 𝒓 = 𝜽 tan−1 𝒚 𝒙 = 𝜽 cot−1 𝒙 𝒚 = 𝜽 cosec−1 𝒓 𝒚 = 𝜽 sec−1 𝒓 𝒙 = 𝜽 Trigonometry Inverse of Ratios x y r θ
- 12. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg tan 𝜽 = sin 𝜽 cos 𝜽 tan2 𝜽 + 1 = sec2 𝜽 sin2 𝜽 + cos2 𝜽 = 1 cosec 𝜽 = 𝟏 sin 𝜽 Trigonometry Identities sec 𝜽 = 𝟏 cos 𝜽 cot 𝜽 = cos 𝜽 sin 𝜽 cot2 𝜽 + 1 = cosec2 𝜽 tan 𝜽 × cot 𝜽 = 1
- 13. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Trigonometry Products 2 cos 𝐴 cos 𝐵 = cos 𝐴 − 𝐵 + cos 𝐴 + 𝐵 2 cos 𝐴 sin 𝐵 = sin 𝐴 + 𝐵 − sin 𝐴 − 𝐵 2 sin 𝐴 cos 𝐵 = sin 𝐴 + 𝐵 + sin 𝐴 − 𝐵 2 sin 𝐴 sin 𝐵 = cos 𝐴 − 𝐵 − cos 𝐴 + 𝐵
- 14. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Trigonometry Sums cos 𝐶 + cos 𝐷 = 2 cos 𝐶 + 𝐷 2 × cos 𝐶 − 𝐷 2 sin 𝐶 − sin 𝐷 = 2 cos 𝐶 + 𝐷 2 × sin 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
- 15. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg tan 𝐴 − 𝐵 = tan 𝐴 − tan 𝐵 1 + tan 𝐴 tan 𝐵 sin 𝐴 − 𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵 cos 𝐴 + 𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵 sin 𝐴 + 𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵 cos 𝐴 − 𝐵 = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 tan 𝐴 + 𝐵 = tan 𝐴 + tan 𝐵 1 − tan 𝐴 tan 𝐵 Trigonometry Compound Angles
- 16. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝑨 = 𝟏 𝟐 𝒓 𝟐 𝜽 𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 = 𝟏𝟖𝟎° 𝒂 𝟐 = 𝒃 𝟐 + 𝒄 𝟐 − 𝟐𝒃𝒄 cos 𝑨 𝒂 sin 𝑨 = 𝒃 sin 𝑩 = 𝒄 sin 𝑪 𝒔 = 𝒓𝜽 𝑨 = 𝟏 𝟐 𝒂𝒃 sin 𝑪 Trigonometry Rules Sine Rule Cosine Rule Area of a triangle Radians Length of arc Area of sector
- 17. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Trigonometry General Solutions 𝐼𝑓 cos 𝜃 = cos 𝛼 𝑡ℎ𝑒𝑛 𝜃 = 2𝑛𝜋 ± 𝛼, 𝑛 ∈ 𝐼 𝐼𝑓 sin 𝜃 = sin 𝛼 𝑡ℎ𝑒𝑛 𝜃 = 𝑛𝜋 + −1 𝑛 𝛼, 𝑛 ∈ 𝐼 𝐼𝑓 tan 𝜃 = tan 𝛼 𝑡ℎ𝑒𝑛 𝜃 = 𝑛𝜋 + 𝛼, 𝑛 ∈ 𝐼 sin(90 − 𝜃) = cos 𝜃 cos 90 − 𝜃 = sin 𝜃 tan 90 − 𝜃 = cot 𝜃 Complementary angles Multiple angles
- 18. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg 𝑃 = 2 𝑙 + 2 𝑏 𝐴 = 𝑙 𝑏 𝐶 = 2𝜋𝑟 𝐴 = 𝜋 𝑟2 Geometry Perimeter AreaShape a b c h l b r 𝑃 = 𝑎 + 𝑏 + 𝑐 𝐴 = 1 2 𝑏 ℎ 𝑠 = 1 2 (𝑎 + 𝑏 + 𝑐) 𝐴 = 𝑠 𝑠 − 𝑎 𝑠 − 𝑏 𝑠 − 𝑐
- 19. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Geometry Surface AreaShape Volume 𝑆 𝐴 = 4𝜋𝑟2 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑉 = 4 3 𝜋𝑟3 𝑆 𝐴 = 2𝜋𝑟ℎ + 2𝜋𝑟2 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠, ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑉 = 𝜋𝑟2 ℎ Sphere Cylinder Cone 𝑆 𝐴 = 𝜋𝑟𝑙 + 𝜋𝑟2 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑙 = 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝑉 = 1 3 𝜋𝑟2ℎ
- 20. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Geometry Surface AreaShape Volume 𝑆 𝐴 = 6𝑙2 𝑙 = 𝑒𝑑𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑉 = 𝑙3 𝑆 𝐴 = 2𝑙𝑏 + 2𝑙ℎ + 2𝑏ℎ 𝑙 = 𝑒𝑑𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑏 = 𝑏𝑟𝑒𝑎𝑑𝑡ℎ ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑉 = 𝑙𝑏ℎ 𝑆 𝐴 = 3𝑙ℎ + 2𝐵𝐴 𝑙 = 𝑏𝑎𝑠𝑒 𝑒𝑑𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ℎ = 𝑝𝑟𝑖𝑠𝑚 ℎ𝑒𝑖𝑔ℎ𝑡 𝐵𝐴 = 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 𝑉 = 𝐵𝐴 ℎ Cube Cuboid
- 21. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Geometry Conic Sections 𝑥2 𝑎2 + 𝑦2 𝑏2 = 1 𝑥 = 𝑎 cos 𝜃 , 𝑦 = 𝑏 sin 𝜃 𝑜𝑟 𝑥 = 𝑎𝑡2 , 𝑦 = 2𝑎𝑡 𝑥 − 𝑎 2 + 𝑦 − 𝑏 2 = 𝑟2 ℎ𝑎𝑠 𝑐𝑒𝑛𝑡𝑟𝑒 𝑎𝑡 𝑎, 𝑏 𝑎𝑛𝑑 𝑟𝑎𝑑𝑖𝑢𝑠 𝑟 𝑥 = 𝑎 + 𝑟 cos 𝜃 𝑦 = (𝑏 + 𝑟 sin 𝜃) AlgebraicShape Parametric Circle Ellipse 𝑥2 𝑎2 − 𝑦2 𝑏2 = 1 𝑥 = 𝑎 sec 𝜃 , 𝑦 = 𝑏 tan 𝜃
- 22. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Geometry Conic Sections 𝑥2 = 4𝑎𝑦 Algebraic EquationShape 𝑎𝑥2 + 𝑏𝑦2 + 2𝑔𝑥 + 2𝑓𝑦 + 2ℎ𝑥𝑦 + 𝑐 = 0 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏, 𝑐, 𝑓, 𝑔, ℎ ∈ 𝑅 Parabola General Equation to a Conic
- 23. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Calculus Limits lim 𝑥→0 sin 𝜃 𝜃 = 1 lim 𝑛→∞ 𝑎 𝑥 𝑛 = 𝑎 Incremental Limits 𝑑𝑦 𝑑𝑥 = 𝑓′ 𝑥 = lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓(𝑥) ℎ
- 24. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Calculus Differentiation 𝐺𝑖𝑣𝑒𝑛 𝑦 = 𝑓 𝑥 𝑎𝑛𝑑 𝑦′ = 𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 𝑓 𝑥 𝑓′ 𝑥 c 0 𝑥 𝑛 𝑛𝑥 𝑛−1 ln 𝑥 1 𝑥 𝑒 𝑎𝑥 𝑎𝑒 𝑎𝑥 𝑎 𝑥 𝑎 𝑥 ln 𝑎 𝑓 𝑥 𝑓′ 𝑥 sin 𝑥 cos 𝑥 cos 𝑥 −sin 𝑥 tan 𝑥 sec2 𝑥 sec 𝑥 sec 𝑥 tan 𝑥 cosec 𝑥 cosec 𝑥 cot 𝑥 cot 𝑥 − cosec2 𝑥
- 25. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Calculus Differentiation Rules 𝐺𝑖𝑣𝑒𝑛 𝑢 = 𝑓 𝑥 𝑎𝑛𝑑 𝑣 = 𝑔 𝑥 𝑢𝑣 ′ = 𝑓′ 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔′ 𝑥 𝑜𝑟 𝑖𝑓 𝑦 = 𝑢𝑣 𝑑𝑦 𝑑𝑥 = 𝑑𝑢 𝑑𝑥 𝑣 + 𝑢 𝑑𝑣 𝑑𝑥 Product Rule Quotient Rule 𝑢 𝑣 ′ = 𝑣𝑢′ − 𝑢𝑣′ 𝑣2 𝑜𝑟 𝑖𝑓 𝑦 = 𝑢/𝑣 𝑑𝑦 𝑑𝑥 = 𝑣 𝑑𝑢 𝑑𝑥 − 𝑢 𝑑𝑣 𝑑𝑥 𝑣2
- 26. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Calculus Differentiation Rules 𝐺𝑖𝑣𝑒𝑛 𝑢 = 𝑔 𝑥 𝑎𝑛𝑑 𝑦 = 𝑓 𝑢 𝑑𝑦 𝑑𝑥 = 𝑑𝑦 𝑑𝑢 . 𝑑𝑢 𝑑𝑥 Chain Rule or Composite Function Rule Gradient 𝑑𝑦 𝑑𝑥 𝑜𝑓 𝑓 𝑥 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝑥1, 𝑦1 = 𝑚
- 27. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Calculus Integration 𝐲 = 𝑓 𝑥 𝒇 𝒙 𝒅𝒙 k 𝑘𝑥 + 𝑐 𝑥 𝑛 𝑥 𝑛+1 𝑛 + 1 + 𝑐 1 𝑥 ln 𝑥 + 𝑐 sin 𝑥 −cos 𝑥 + 𝑐 cos 𝑥 sin 𝑥 + 𝑐
- 28. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Calculus Integration Rules 𝑎 𝑏 𝑓′ 𝑥 𝑑𝑥 = 𝑓 𝑏 − 𝑓 𝑎 𝑓 𝑥 𝑔 𝑥 𝑑𝑥 = 𝑓(𝑥) 𝑔 𝑥 𝑑𝑥 − 𝑔(𝑥) 𝑑𝑥 𝑑𝑓(𝑥) 𝑑𝑥 Volume Integral 𝑉 = 𝜋 𝑎 𝑏 𝑦2 𝑑𝑥
- 29. FORMULAE FOR STUDENTS MATHEMATICS ©2014 MDR www.learningforknowledge.com/glg Author: M. D. Raghu Email: glg@learningforknowledge.com Ganit Learning Guides Thank you Design-Layout: www.narayanasoftware.com For exemplars and proofs of formulae Visit: www.learningforknowledge.com/glg