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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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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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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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 =


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-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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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


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 =


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 …... E R K HIGHER SECONDARY SCHOOL-ERUMIYAMPATTI SOME IMPORTANT...

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