E R K HIGHER SECONDARY SCHOOL …... E R K HIGHER SECONDARY SCHOOL-ERUMIYAMPATTI SOME IMPORTANT...
Transcript of E R K HIGHER SECONDARY SCHOOL …... E R K HIGHER SECONDARY SCHOOL-ERUMIYAMPATTI SOME IMPORTANT...
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E R K HIGHER SECONDARY SCHOOL-ERUMIYAMPATTISOME IMPORTANT RESULTS IN ANALYTICAL GEOMETRY
Y
Fi
Yû[YûW gutisak; ePs;tl;lk; mjpgutisak; nrt;tf mjpgutisak;
1. fhh;Brpad; rkd;ghL y2 = 4ax 12
2
2
2
b
y
a
x1
2
2
2
2
b
y
a
xxy = c2
2. JizayF rkd;ghL x = at2 ; y = 2at
,q;F t- JizayF
x = a cos θ ; y = b sin θ
,q;F θ-JizayF
x = a sec θ ; y = b tanθ
,q;F θ-JizayF
x = ct ; y =
,q;F t-JizayF
3. (x1, y1) Gs;spaplj;J
njhL Nfhl;bd; rkd;ghL yy1 = 2a(x + x1) 1
2
1
2
1
b
yy
a
xx1
2
1
2
1
b
yy
a
xxxy1 + yx1 = 2c2
4. t (m) Gs;spaplj;J
njhLNfhl;bd; rkd;ghL yt = x + at2
1sincos
b
y
a
x 1
tansec
b
y
a
x
x + yt2 = 2ct.
(OR)
5. (x1,y1) Gs;spaplj;J
nrq;Nfhl;bd; rkd;ghL x y1 + 2ay = x1y1 + 2ay1
22
1
2
1
2
bay
yb
x
xa
22
1
2
1
2
bay
yb
x
xa
6. t (m) Gs;spaplj;J
nrq;Nfhl;bd; rkd;ghL 22
sincosba
byax
22
tansecba
byax
(OR)
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.inY
Fi
Yû[YûW gutisak; ePs;tl;lk; mjpgutisak; nrt;tf mjpgutisak;
7. y = mx + c njhLNfhlhf miktjw;fhd epge;jid
C = c2 = a2m2 + b2 c2 = a2m2 - b2 -
8. njhLk; Gs;sp
m
a
m
a 2,
2
c
b
c
ma22
,
c
b
c
ma22
, -
9. FkR JÚ njhLNfhl;bd; rkd;ghL
-
10.
lx + my + n = 0 vd;w NfhL
njhLNfhlhfmika epge;jid
nlam 2 22222
nmbla 22222
nmbla 22
4 nlmc
11. 0 nmylx vd;w
NfhL nrq;Nfhlhfmikaepge;jid 02
223 nmalmal 2
222
2
2
2
2)(
n
ba
m
b
l
a
2
222
2
2
2
2)(
n
ba
m
b
l
a
-
12.
xU Ftpaj;jpypUe;J xU njhLNfhl;bw;F
tiuag;gLk; nrq;Fj;Jf; Nfhl;bd; mbapd;
epakg;ghij
0x222
ayx
(ÕûQ YhPm)
222ayx
(ÕûQ YhPm)
-
13. nrq;Fj;J njhLNfhLfs;
ntl;bf;nfhs;Sk;Gs;spapd; epakg;ghij
x = − a (BVdÏYûW)
2222bayx
(BVdÏYhPm)
2222bayx
(BVdÏYhPm)
-
14. tiuag;gLk;
njhLNfhLfspd; vz;zpf;if
2 2 2 2
15. tiuag;gLk;
nrq;NfhLfspd; vz;zpf;if
3 4 4 4
16.
,af;Ftiuapd; ve;j xU Gs;spapypUe;Jk;
tiuag;gLk; njhLNfhLfspd;
njhLehz;
Ftpak; (topahf nry;Yk;)
xj;j Ftpak; (topahf nry;Yk;)
xj;j Ftpak; (topahf nry;Yk;) -
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E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -1
IMPORTANT DIFFERENTIATION & INTEGRATION FORMULA
S:NO DIFFERENTIATION INTEGRATION
1. ( )= n : n ∫ dx= +c : where
2. (x) = 1 ∫ = x+ c
3. (log x) = ∫ dx = log x + c
4. (k) = 0 : where k is an constant ∫
5. ( ) = ∫
6. )= ∫
7. ) ∫
8. ) ∫
9. ) ∫
10. ) ∫
11. ) x ∫ )
12. ) ∫ )
13. ) ∫ )
14. ) ∫ )
15. - ∫
16. - ∫
17. - ∫
18. - ∫
19. ) +v ∫ ∫
20. ) = -
21. )√
∫√
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E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -2
22. )√
∫√
23. ) ∫
24. ) √
∫ √
25. ) √
∫ √
26. ) ∫
27. ∫ )
) )
28. ∫ )
√ )√ )
29. ∫ ) + c
30. ∫ ) + c
31. ∫ ) + c
32. ∫√
[x+√ ] + c
33. ∫√
[x+√ ] + c
34. ∫√
) + c
35. ∫√ √ [x+√ ] + c
36. ∫√ √ [x+√ ] + c
37. ∫√ √ ) + c
38. ∫ sinbx + c
39. ∫ cosbx + c
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E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3
IMPORTANT TRIGONOMETRICAL FORMULA
TRIGONOMETRICAL IDENTITIES:
sin =
cosec =
cos =
sec =
tan = (OR) tan =
cot = (OR) cot =
sin2 + cos2 = 1
sin2 = 1- cos2
cos2 =1- sin2
1+tan2 = sec2
sec2 - tan2 = 1
tan2 = sec2 -1
1+cot2 = cosec2
cosec2 - cot2 = 1
cot2 = cosec2 - 1
TRIGONOMETRICAL TABLES:
O
Sin 0 √
√ 1 0 -1 0
Cos 1 √
√ 0 -1 0 1
tan 0 √
1 √ 0 0
ASTC RULE:
I Quadrant A All the trigonometric functions are positive
II Quadrant S sin & cosec only positive, remains are negative
III Quadrant T tan & cot only positive, remains are negative
IV Quadrant C cos & sec only positive, remains are negative
T-Ratios of (90 ) & (270 ) sin cos ; tan cot ; cosec sec
T-Ratios of (180 ) & (360 ) No change in trigonometric functions
SOME SPECIAL PROPERTIES OF TRIGONOMETRICAL FUNCTIONS:
Even Function: Put x=- x in f(x), then we get f(-x)= f(x)
Odd Function: Put x=- x in f(x), then we get f(-x)= -f(x)
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E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -4
S.NO Compound Angles for: sin(A B) , cos(A B) & tan(A B) 1. Sin(A+B) = sinAcosB + cosAsinB
2. Sin(A-B) = sinAcosB - cosAsinB
3. Cos(A+B) = cosAcosB – sinAsinB
4. Cos(A-B) = cosAcosB + sinAsinB
5. Sin(A+B) + Sin(A-B) = 2 sinAcosB
6. Sin(A+B) - Sin(A-B) = 2 cosAsinB
7. Cos(A+B)+ Cos(A-B)= 2 cosAcosB
8. Cos(A+B)- Cos(A-B)= -2 sinAsinB (OR) Cos(A-B)- Cos(A+B)= 2 sinAsinB
9. tan(A+B) =
10. tan(A-B) =
S.NO Multiple Angles for: sin2A & cos2A
1. Sin2A = 2SinAcosA (OR) sinA = 2sin cos
2. Cos2A = cos2A- sin2A (OR) cosA = cos2 - sin2
3. Cos2A = 2 cos2A – 1 (OR) cos2A =
4. Cos2A = 1 - 2 sin2A (OR) sin2A =
S.NO Multiple Angles for: sin3A & cos3A
1. Sin3A = 3sinA – 4sin3A
2. Cos3A = 4cos3A – 3cosA
S.NO Properties of Inverse trigonometric functions
1. + = √ + √ ]
2. - = √ - √ ]
3. + = ⌈ ⌉
4. - = ⌈ ⌉
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E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -1
E R K HIGHER SECONDARY SCHOOL-ERUMIYAMPA
SOME IMPORTANT TRIGONOMETRICAL FORMULA’S
TRIGONOMETRICAL IDENTITIES:
sin =
cosec =
sin =
cosec =
cos =
sec =
cos =
sec =
tan =
cot =
tan = (OR) tan =
cot = (OR) cot =
sin2 + cos2 = 1
sin2 = 1- cos2
cos2 = 1- sin2
1+tan2 = sec2
sec2 - tan2 = 1
tan2 = sec2 -1
1+cot2 = cosec2
cosec2 - cot2 = 1
cot2 = cosec2 - 1
TRIGONOMETRICAL TABLES:
O
sin 0 √
√ 1 0 -1 0
cos 1 √
√ 0 -1 0 1
tan 0 √
1 √ 0 0
ASTC RULE:
I Quadrant (90 A All the trigonometric functions are positive
II Quadrant ( (OR) (180 ) S sin & cosec only positive, remains are negative
III Quadrant ( (OR) (270 ) T tan & cot only positive, remains are negative
IV Quadrant ( (OR) (360 ) C cos & sec only positive, remains are negative
T-Ratios of (90 ) (OR) (270 ) sin cos ; tan cot ; cosec sec (T-functions Changes)
T-Ratios of (180 ) (OR) (360 ) No change in trigonometric functions
SOME SPECIAL PROPERTIES OF TRIGONOMETRICAL FUNCTIONS:
Even Function: Replace x =- x in f(x), then we get f(-x) = f(x)
Odd Function: Replace x =- x in f(x), then we get f(-x) = -f(x)
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E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -2
S.NO Compound Angles for: sin(A B) , cos(A B) & tan(A B)
1. sin(A+B) = sinA cosB + cosA sinB
2. sin(A-B) = sinA cosB – cosA sinB
3. cos(A+B) = cosA cosB – sinA sinB
4. cos(A-B) = cosA cosB + sinA sinB
5. sin(A+B) + Sin(A-B) = 2 sinA cosB
6. sin(A+B) - Sin(A-B) = 2 cosA sinB
7. cos(A+B)+ Cos(A-B) = 2 cosA cosB
8. cos(A+B)- Cos(A-B) = -2 sinA sinB (OR) cos(A-B)- cos(A+B) = 2 sinA sinB
S.NO Multiple Angles for: sin2A & cos2A
1. sin2A = 2SinA cosA (OR) sinA = 2sin cos
2. cos2A = cos2A- sin2A (OR) cosA = cos2 - sin2
3. cos2A = 2 cos2A – 1 (OR) cos2A =
4. cos2A = 1 - 2 sin2A (OR) sin2A =
S.NO Multiple Angles for: sin3A & cos3A
1. sin3A = 3sinA – 4sin3A (OR) sin3A =
2. cos3A = 4cos3A – 3cosA (OR) cos3A =
S.NO Properties of Inverse trigonometric functions:
1. + = √ + √ ]
2. - = √ - √ ]
3. + = ⌈ ⌉
4. - = ⌈ ⌉
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E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3
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