E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

10
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 y 2 = 4ax 1 2 2 2 2 b y a x 1 2 2 2 2 b y a x xy = c 2 2. JizayF rkd;ghL x = at 2 ; 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 = 2c 2 4. t (m) Gs;spaplj;J njhLNfhl;bd; rkd;ghL yt = x + at 2 1 sin cos b y a x 1 tan sec b y a x x + yt 2 = 2ct. (OR) 5. (x1,y1) Gs;spaplj;J nrq; Nfhl;bd; rkd;ghL x y1 + 2ay = x1y1 + 2ay1 2 2 1 2 1 2 b a y y b x x a 2 2 1 2 1 2 b a y y b x x a 6. t (m) Gs;spaplj;J nrq; Nfhl;bd; rkd;ghL 2 2 sin cos b a by ax 2 2 tan sec b a by ax (OR) www.Padasalai.Net www.TrbTnpsc.com

Transcript of E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

Page 1: 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)

www.Padasalai.Net

www.TrbTnpsc.com

Page 2: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

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;) -

www.Padasalai.Net

www.TrbTnpsc.com

Page 3: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

www.Padasalai.Net

www.TrbTnpsc.com

Page 4: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

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.

)

√ ∫

www.Padasalai.Net

www.TrbTnpsc.com

Page 5: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

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

www.Padasalai.Net

www.TrbTnpsc.com

Page 6: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

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)

www.Padasalai.Net

www.TrbTnpsc.com

Page 7: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

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.

- = ⌈

www.Padasalai.Net

www.TrbTnpsc.com

Page 8: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

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)

www.Padasalai.Net

www.TrbTnpsc.com

Page 9: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

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.

- = ⌈

www.Padasalai.Net

www.TrbTnpsc.com

Page 10: E R K HIGHER SECONDARY SCHOOL …...E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

E R K HIGHER SECONDARY SCHOOL- ERUMIYAMPATTI PAGE -3

www.Padasalai.Net

www.TrbTnpsc.com