fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM/IISc, JEST, TIFR and GRE in PHYSICAL SCIENCES Basic Mathematics Formula Sheet for Physical Sciences Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 1 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Basic Mathematics Formula Sheet for Physical Sciences 1. Trigonometry…………………………………………………………………… (3-9) 1.1 Trigonometrical Ratios and Identities………………………………………...… (3-7) 1.2 Inverse Circular Functions………………………………………...……………. (8-9) 2. Differential and integral Calculus………………………………………….… (10-20) 2.1 Differentiation …………...……………. ……………………………………... (10-12) 2.2 Limits …………………………………………………………………..………(13-14) 2.3 Tangents and Normal……………………………..…………………………….(15-16) 2.4 Maxima and Minima ……………………………………………………………..(16) 2.5 Integration……………………………………………………………………....(17-19) 2.5.1 Gamma integral……………………………………………………..(19) 3. Differential Equations………………………………………………….……....(20-22) 4. Vectors……………….…….................................................................................(23-25) 5. Algebra……………….…….................................................................................(26-32) 5.1 Theory of Quadratic equations…………………………………………………...(26) 5.2 Logarithms………………………………………………………………………..(27) 5.3 Permutations and Combinations……………………………………………......(28-29) 5.4 Binomial Theorem…………………………………………………......................(30) 5.5 Determinants……………………………………………....................................(31-32) 6. Conic Section……………….……..........................................................................(33) 7. Probability……………….……...........................................................................(34-35) Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 2 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1. Trigonometry 1.1 Trigonometrical Ratios and Identities 1. sin2cos21 2. sec21tan2 sin 3. cosec21cot2 4. tan cos cos 1 5. cot 6. sin sin cosec 1 1 7. cos 8.tan sec cot Addition and Subtraction Formulae For any two angles A and B     1. Sin AB sinAcosBcosAsinB 2. Sin AB sinAcosBcosAsinB 3. cosABcosAcosBsin AsinB 4. cosABcosAcosBsin AsinB tanAtanB tan AtanB 5. tanAB 6. tanAB 1tanA.tanB 1tan A.tanB Double Angle Formulae 1. sin22sincos, 2. cos2cos2sin212sin2  2cos21 2tan 3. tan2 1tan2 Triple angle Formulae 1. sin33sin4sin3 2. cos34cos33cos 3tantan3 3. tan3 13tan2 Trigonometric Ratios of θ/2       1. sin 2sin cos , 2. coscos2 sin2  2cos2 12sin2 2 2 2 2 2 2  2tan 2 3. tan  1tan2 2 Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 3 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Formulae for sin2θ & cos2θ in terms of tanθ 2tan 1tan2 1. sin2 2. cos2 1tan2 1tan2 Formulae for sinθ & cosθ in terms of tanθ/2   2tan 1tan2 2 2 1. sin 2. cos   1tan2 1tan2 2 2 Transformation of sum/differences into Products C D C D 1. sinC sinD 2sin cos   2   2  C D C D 2. sinC sinD  2cos sin   2   2  CD CD 3. cosC cosD 2cos cos   2   2  C D C D CD DC 4. cosCcosD 2sin sin  2sin sin   2   2   2   2  Transformations of Products into sum/difference 1. 2SinAcosB  Sin(AB)Sin(AB) 2. 2cosAsinB  Sin(AB)Sin(AB) 3. 2cosAcosB cos(AB)cos(AB) 4. 2sin AsinB cos(AB)cos(AB) Trigonometric Ratios of (-θ)     1. sin   sin 2. cos  cos 3. tan tan 4. cot     5. sec   cot 6. cosec   cosec Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 4 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES   Trigonometric Ratio of : (All Positive)  2      1. cos sin 2. sin cos  2  2      3. tan cot 4. cot  tan  2   2      5. cosec sec 6. sec cosec  2   2    Trigonometric Ratio of  :( Onlysin and cosec is Positive)  2      1. cos  sin 2. sin cos  2  2      3. tan cot 4. cot tan  2   2      5. cosec sec 6. sec  cosec  2   2    Trigonometric Ratios of  :( Onlysin and cosec is Positive) 1. cos cos 2. sinsin     3. tan  tan 4. cot cot     5. cosec cosec 6. sec sec Trigonometric Ratios of :( Onlytan and cot is Positive)     1. cos  cos 2. sin  sin 3. tan tan 4. cotcot 5. cosec cosec 6. sec sec Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 5 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 3  Trigonometric Ratio of  :( Onlytan and cot is Positive)  2  3  3  1. cos sin 2. sin  cos  2   2  3  3  3. tan cot 4. cot  tan  2   2  3  3  5. cosec sec 6. sec cosec  2   2  3  Trigonometric Ratio of  :( Onlycos and sec is Positive)  2  3  3  1. cos sin 2. sin cos  2   2  3  3  3. tan  cot 4. cot tan  2   2  3  3  5. cosec  sec 6. sec cosec  2   2    Trigonometric Ratios of 2 :( Onlycos and sec is Positive) 1. cos2cos 2. sin2 sin     3. tan 2  tan 4. cot 2  cot     5. cosec 2 cosec 6. sec 2 sec Trigonometric Ratios of2: (All Positive)     1. cos 2 cos 2. sin 2 sin 3. tan2 tan 4. cot2cot 5. cosec2cosec 6. sec2sec Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 6 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Short-cut method to remember the Trigonometric ratios n  1. sin  sin  2  n  2. cos cos when n is an even integer  2  n  n  3. tan tan 4. sin  cos  2   2  n  5. cos sin when n is an odd integer  2  n  6. tan cot  2  θ 0o 30o 45o 60o 90o 120o 135o 150o 180o 270o 360o sin θ 0 1 1 1 1 1 0 -1 0 3 3 2 2 2 2 2 2 cos θ 1 3 1 1 0 1 1 3 -1 0 1    2 2 2 2 2 2 tan θ 0 1 1 3 ∞  3 -1 1 0 ∞ 0  3 3 cot θ ∞ 3 1 1 0 1 -1  3 ∞ 0 ∞ 3 3 sec θ 1 2 2 2 ∞ -2  2 2 -1 ∞ 1 3 3 cosec θ ∞ 2 2 2 1 2 2 2 ∞ -1 ∞ 3 3 Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 7 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1.2 Inverse Circular Functions 1. sin1sinx x 2. cos1cosx x 3. tan1tanx x 4. cot1cotx x 5. sec1secx x 6. cosec1cosec x     7. sin sin1 x  x 8. coscos1 x  x     9. secsec1 x  x 10. cosec cosec1x  x 1 1 11. sin1  cosec1x 12. cos1  sec1 x x x 1 1 13. tan1 cot1 x 14. cot1  tan1 x x x 1 1 15. sec1 cos1 x 16. cosec1 sin1 x x x 17. sin1x sin1 x 18. cos1xcos1 x  19. tan1x tan1 x 19. sin1 xcos1 x  2   20. tan1 xcot1 x  21. sec1 xcosec1x  2 2 22. sin1 1 x2  cos1 x 23. cos1 1 x2 sin1 x 24. tan1 x2 1 sec1 x 25. cot1 x2 1 cosec1x 26. sec1 1x2  tan1 x 27. cosec1 1 x2 cot1 x     28. sin1 2x 1x2 2sin1 x 29. sin1 3x4x3 3sin1 x    2x  30. cos1 4x3 3x 3cos1 x 31. tan1  2tan1 x 1x2  3xx3   x y  32. tan1  3tan1 x 33. tan1   tan1 xtan1 y     13x2  1xy  x y  34. tan1  tan1 xtan1 y   1 xy Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 8 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES x3 x5 Some Important Expansions: 1. sinx  x  ......... 3! 5! x3 x5 x2 x4 2. sinhx  x  ......... 3. cosx 1  ......... 3! 5! 2! 4! x2 x4 1 2 4. coshx 1  ......... 5. tanx  x x3  x5 ......... 2! 4! 3 15 x2 x3 x2 x3 6. ex 1x  ......... 7. ex 1x  ......... 2! 3! 2! 3! Some useful substitutions:- Expressions Substitution Formula Result 3x4x3 x sin 3sin4sin3 Sin3θ 4x3 3x x  cos 4cos33cos cos3θ 3xx3 x = tan θ 3tantan3 tan3θ 13x2 13tan2 2x x = tan θ 2tan sin2θ 1x2 1tan2 1x2 x = tan θ 1tan2 cos2θ 1x2 1tan2 2x x = tan θ 2tan tan2θ 1x2 1tan2 12x2 x = sin θ 12sin2 cos2θ 2x2 1 x = cos θ 2cos21 cos2θ 1x2 x = sin θ 1sin2 cos2θ 1x2 x = cos θ 1cos2 sin2θ x2 1 x = sec θ sec21 tan2θ x2 1 x = cosec θ cosec21 cot2θ 1x2 x = tan θ 1tan2 sec2θ 1x2 x = cot θ 1cot2 cosec2θ Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 9 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2. Differential and integral Calculus 2.1 Differentiation         f xh  f x f ah  f a 1. fxlim 2. f'alim h0 h h0 h dy y d   3.  lim 4. k 0; k is constant function dx x0x dx d d   1 5. x1 6. x  dx dx 2 x d  1  n d   7.    8. xn  nxn1;nN dxxn  xn1 dx d 1 1 d 9.   10. sinxcosx dxx x2 dx d d 11. cosx sinx 12. tanxsec2 x dx dx d d     13. secx secx.tanx 14. cosecx cosecx.cotx dx dx 15. d ax ax loga;a 0,a 1 16. d exex dx dx 17. d log x 1 18. d sin1 x 1 ; 1 x1 dx x dy 1x2 d   1 d   1 19. cos1 x  ;1 x1 20. tan1 x  ; xR dx 1x2 dx 1x2 d   1 d   1 21. cot1 x  ; xR 22. sec1  : x 1 dx 1 x2 dx x x2 1 d   1 23. cosec1x  ; x 1 dx x x2 1 Head office Branch office fiziks, H.No. 23, G.F, Jia Sarai, Anand Institute of Mathematics, Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT Phone: 011-26865455/+91-9871145498 Hauz Khas, New Delhi-16 Website: www.physicsbyfiziks.com Email: [email protected] 10