E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3
Transcript of E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3
E R K HIGHER SECONDARY SCHOOL-ERUMIYAMPATTI SOME 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
x 1
2
2
2
2
b
y
a
x xy = 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
xx 1
2
1
2
1
b
yy
a
xx xy1 + yx1 = 2c2
4.
t (m) Gs;spaplj;J
njhLNfhl;bd; rkd;ghL yt = x + at2
1sincos
b
y
a
x
1tansec
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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Y
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
22222nmbla
22222
nmbla
22
4 nlmc
11.
0 nmylx vd;w NfhL nrq;Nfhlhfmika
epge;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
0x
222ayx
(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
E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI 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-ERUMIYAMPATTI
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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