Mathematics Formula Sheet - CSIR NET Physics

35 Pages·2013·0.88 MB·English

fiziks
Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

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Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM/IISc,
JEST, TIFR and GRE in PHYSICAL SCIENCES

Basic Mathematics Formula Sheet for
Physical Sciences

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Near IIT, Hauz Khas, New Delhi-16 28-B/6, Jia Sarai, Near IIT

Phone: [phone] 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)

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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

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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

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 
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

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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

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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

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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

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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

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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

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