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rigonometry
Sin'+ (^) cos*e 1 1tcote cosec' (^14) tan'6 sec Sin (^) 20:2sinGcos
cos202cos 9-
1-2sin" Cos'e- sin o Cos' (^) 1+cos 2
Sin'9 1-cos26e
Sin& (^) 2sinco
cose cos9 -sin
Zcos-
1-2sin
cosgtcos 2 si g^ -cos
Sin 30 BsinG- 4 sin'e
co6e (^) hcose 3cos Sin 29 2tan 1+ 4an e
co s20= 1-tan
1+Han
tan 29:2lanG
1-tane
.2sinAcos6 in^ (A+)+sin(A-B) 2osAsin B^ Sin^ A*B)^ -sin^ (A-B) 2s0sA cosB: cos^ (A4 G)^ +cos (^) (A-6) 2in A^ sin^ 6:^ Cos^ A-8)-cos(A+^ 8) Sin CAt8)-^ sin Acos^6 t^ cosAsin cos (A8)^ co5AcosB^ SinAsin^6
Lan (At8) LanAlanA^ lan^ G
tanB
LIC (^) 5ino NURAnc Cosec
Sin (-e)^ Sin
co (-e)^ cos
ton (-6)-lon
cos
ee
tonO
cot O SinA (^) sin :^2 sin Acos(A
inA- sinG
2cos(A1i
cosA tcos
2co (^) (A (^) cos
-e
cos (^) A-cos8 (^) -2sin(A8Sin(A
45 Go
/6 (^) d/3 T/ SinG (^) O (^) V2 V cose (^) S/ VS
tanG
coseg 2
seo J2 N-D
cote ND O
Sinml O (^) cos nT (-1)" sin (2-)IL-(-) cos (^) (2n-)T
Atve
T
+vean^ Cos
1Si (^) e)-tcos sin (^) -e):tcos o
Cos(-to+sin
sin (11-o)^ +sin
sinT+0)-SinG
Cos (2TT-) +cos
Cos (2T-+)^
+cose 1cos(e)-si
2LIC utwsuA CORPOR
Intgvator
foc)
Skdx nt logc 2Jz
Sedx Sadx
ec
loga )Sinzdz^ - cosx +c 9Jcosz dz secx Hanz^ dx
Sine tc
Sec oc^ tc
1o) J cosecz colz^ dz^ -Cosecetc
sece dx^ tan^ t^ c
- (^) Jcosece dz^ - cotet^ c
tan4c oR^ -cot^ tc
L Sinde oR^ -cos ztc sec+e oR^ cosec^ +c
19Lane dx log^ Isecel
4
19cot dz^ log^ Sin^ l+c
)secoedx^ loqlsecx^ + Hane| 4 c
1)cosecx dx^ o^ eosecx - cot el4^ c
sn (te
a
(^9) JJ4a logl +Jza|
(^2) dx log (^) |T +^ JLltc
cosbx dx^ eacosbx 4 bsinbx) a' feesanbx^ dr^ e a'4b^ aunb^ - bcosbr)
(LIATE)
Le:bnitz (^) ule (^) (only AT ov AE) uv dr^ uv-uv,+uVs..
LAPLACE
Def" ofLT.
L[fe- (^) fC) 4)Jt
Standard tovmulae
-at s4a eat Sinat
cos at
a Sinhat
coshat
(Tf n^ is^ tve ineger)
t htnr (If^ n^ is^ tve^ acanad S
cos ht
sin ht
Type- (Scaling prop y) T L)-^ S^ (6) then,
*INVERSE LAPLACE
Standovd fomola^ e
k
(.nis^ tve^ integer)
Cn-1) ((C^ niachonal) T
6ta e Sinat Cosat
sinhat coshat
Type 1- Problems based on^ standavd funethan*^ fovmulae.
Type2 (1st^ shithng^ theorem)
IP LLF(s)): P(t) then, L'LF(s-) e^ ft) L"IFCsto)^ ef() ype 3-Pavtial^ factron erpansion)
NOTE Tf^ degvee^ of^ denomin oy^ is^ always^ gvealev^ tha
degree otnumerater^ hen^ we^ apply^ pavtial
faction methe^ d.
Not rep eakd^ (xtDx-2)(c t)
Pepeakd guadrahe (^) 2)(o+3) A:^43
pe5ASubsiluhon method
P s-^ P&^ sing PFE
Tope (^3) Tnspeclion method 1T (MT 4T
ye:Tnvese LT^ vsiy wles^ of^ LT L[EF). (^) (-1)F($) ds
taking 1LT^ bs
tfE) -L"Ft)
F -t^ fa)
yee 5:^ Conuolulion Theorem T (^) 'F6)J:St) && 'L6S)J 4) then (^) L LF(S) G(] PcgCt-u)du
whe re, fCu)- LfCt)J..
gt-u) Lg (^) C)Jt-u
Type 62ndshif ing heorem
Lea F(s))^ :f(t-a)H(t-a)
FouvierSevies
cosO (^1) sinO O
cos 0
cosS^ O
Sin 2 1
Sin 3 1
Sin 5 1
CoST1: 1 sinTT=O CosZ-1 Sin^2 Oo Cos3T=-1 Sin^311 O
cosn T^ =(-1)"^ Cos^ 2n)^6 *Sinn1^0
*Cos21=^1 *sin^ 2nT^ O^ *sinl2n-)T^ =(-)"
k cos(2n-)7^ cos^ (2nt1)T^ =1^ Sin^ (2n-1)T=^ sin(20+DTI^ =^ o
tINSURANECORPOATON
-,) The (^) fouvie of^ ()^ in^ the^ intewal^ ,)
r)a+a.to(n1d tbSinn
twhere
)in (sT)
ore 3 Even^ &^ Odd^ Function
Even Fonction
The (^) functjon f(oc)^ in^ ,)s said^ do^ be an (^) even (^) Fonction f (^) f-x)-f(x) or (^) an (^) even funchon, f)dz =^2 fd
Od Function The (^) fonetion f(z)^ in (-1,)is^ sad^ to^ be an odd (^) tonction if^ (f-z)=-f()) Far an odd^ funcetion, f)d O
KFovriev Sevies ov an^ even^ unction^ in^ (-8,I)
wher, a, 2'|fCx)^ dx
a (^) 2f)cos)x
Fopier series^
tv an^ odo^ tunclion^ in^ (-,2)
f(x) z
b (^) sin
ohere (^) bJ)sin()d
Type - HalF^ range cosine^ and^ sine^ serie s-
Half (^) Yange cosine^ senes
The half vange cosine sevies ot^ fx)^ in^ the
intewa CO, 1) is
()O+acosn
wheve a.
)co(11)
Hall range sine^ seies
The halF vange sine^ series^ o^ oc)^
in he
inlerval (^) (o,I) is PCx) b. (^) sin ( hoe,b Pt)sin)
COMPLEX (^) VARIABLES
OAnalytie (^) function
Carlesian fonm
u
Uz (^) V (^) Uy-VN then (^) funcion is (^) analy hic. Plar (^) fovm(z: ve")
du
Havmonic funcion
The fonction^ in^ (x,y) is^ said^ to^ be
havmonic tunction^ tisatisties^ aplace
equation i-e,
y
Spea one eovvelalian eoe\cient R 1- 6 E| nn where, d dilerence^ between and^ y anled (^) velues NOTE S^ T the^ nts^ are^ cpen^ ed, hen
adjusthng m.(m-1 12 ilbe^ added^ inle
Summ ation (^) d
whe, m, is vant^ epeu ed m^ tims
Theefot
R 1-^ 6SJ.4^ mi(m1) n(n-1)
Line of Regression
Line of^ regression of y on
(^9) by (x-z)
where, byx regression coefrei ent
by
Line
o egression of^ ony
where, (^) bxy >^ regression coetticien^ t b
6xb bxb
Tf &g are^ integers, hen
by -T(y-y)
Tf (^) yare^ trachiona then benUv-(Su.)(Sv.) nsu-(Eu 6nSu.v, n -(EuX£v:)
Ev.-(Ev.)
Corve iHing
Ast (^) Degre e Staiqht lin^ e
Zne Deg ree Guedvalic^ pavabole
Shaigh line-
Let eq" ol^ sthaight line^ is
gau 6 where (^) CCa
laina summation^ on^ both^ side s, y aSu.t^ nb- Moltply by u&^ Son both^ sides, Suy. a^ Su,'+bEu.^ - 2Juadvatie (^) parabola Let (^) eq eqvadvai pavabola is, 9 y^ ar4ar4^ 6c^ tPavabola^ c^
i
au+bu: 4 c where (^) U; (^) Ci- (^) a
1akinq summation^ on^ both sides,
Sy. (^) a2u+Lzu; (^4) nc Mutply by u &^ zon^ both^ sides, gi (^) a£utbEu+c2u (^2)
Mutiply by u:^ &^ on^ both sides,
SuPy, a£u:'+^ bzu+ (^) cEu
