Formulas , formulae for class xi math

Formulas , formulae for class xi math

1. P a g e |1 NAME:-____________________ CLASS/SEC:- __________


MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)

FORMULAS CH APTER:-10
CLASS:-XI
EXERCISE:-10.1
1:- α+β+� =180
0
2:- 90
0
-α =β
3:- 90
0
-β =α
NOTE:- At 90° AND 270° THE TRIGONOMETRIC FUNCTIONS ARE CHANGED INTO THEIR
CO-FUNCTIONS:
i.e:- Cosθ Sinθ
also:- Tanθ Cotθ
and:- Secθ Cosecθ
WHILE:- AT 180° AND 360° THE TRIGONOMETRIC FUNCTIONS REMAINS SAME.

EXERCISE:-10.2
4:- Sin(α+β)= sin(α)cos(β) + cos(α)sin(β)
5:- Sin(α-β)= sin (α)cos(β) – cos(α)sin(β)
6:- Cos(α+β)= cos(α)cos(β) – sin(α)sin(β)
7:- Cos(α-β)= cos(α)cos(β) + sin(α)sin(β)
8:- Tan(α+β)=
tan�+tan⁡�
1−tan�⁡tan⁡�

9:- Tan(α-β)=⁡
tan�−tan⁡�
1+tan�⁡tan⁡�

10:- Cot(α+β)=⁡
cot�cot�−1
cot�+cot⁡�

11:- Cot(α-β)=⁡
cot�cot�+1
cot�−cot⁡�



EXERCISE:-10.3

2. P a g e |2

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
12:- Sin2α =2SinαCosα
13:- Cos2α = Cos
2
α-Sin
2
α or :- 2cos
2
α -1 or :- 1-2sin
2
α
14:- Tan2α =⁡
2tan⁡α
1−tan
2
�

15:- Sin
α
2
= √
1−����
2

16:- Cos
α
2
= √
1+����
2

17:- Tan
α
2
=√
1−����
1+����

18:- Sin3α = 3Sinα-4Sin
3
α
19:- Cos3α = 4Cos
3
α-3Cosα
20:- Tan3α =⁡
3����−tan
3
�
1−3tan
2
�

21:-Cos2Ø =⁡
1−tan
2
Ø
1+tan
2
Ø

22:-Sin2Ø =⁡
2���Ø
1+tan
2
Ø

EXERCISE:-10.4
PRODUCT TO SUM
23:- 2SinαCosβ =Sin(α+β)+Sin(α-β)
24:- 2CosαSinβ = Sin(α+β)-Sin(α-β)
25:- 2CosαCosβ =Cos(α+β)+Cos(α-β)
26:- -2SinαSinβ =Cos(α+β)-Cos(α-β)
SUM TOPRODUCT
27:- Sin(P)+Sin(Q) =2Sin(
�+�
2
) Cos(
�−�
2
)
28:- Sin(P)-Sin(Q) =2Cos(
�+�
2
) Sin(
�−�
2
)
29:-Cos(P)+Cos(Q) =2Cos(
�+�
2
) Cos(
�−�
2
)
30:- Cos(P)-Cos(Q) = -2Sin(
�+�
2
) Sin(
�−�
2
)

3. P a g e |3

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
FORMULAS CH APTER:-11
CLASS:-XI

-:FOR DOMAIN AND RANGE OF TRIGNOMETRICFUNCTIONS:-

-:PERIODS OF TRIGONOMETRIC FUNCTIONS :-
1:- 2π is the period of Cosθ.
2:- 2π is the period of Sinθ.
3:- 2π is the period of Cosecθ.
4:- 2π is the period of Secθ.
5:- π is the period of Tanθ.
6:- π is the period of Cotθ.
NOTE: [ π IS THE ONLY PERIODS OF Tanθ AND Cotθ. WHILE 2π IS THE PERIODS
OF ALL REMAINING TRIGONOMETRIC FUNCTIONS.]

4. P a g e |4

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
FORMULAS CH APTER:-12
CLASS:-XI
EXERCISE 12.4
LAW OF SINES:- USED WHEN 2 ANGLES & ONE SIDE OR 2 SIDES & 1 ANGLE ARE GIVEN:
1.
�
????????????��
=
�
????????????��

2.⁡
�
????????????��
=
�
????????????��

3.⁡
�
????????????��
=
�
????????????��

4.
�
????????????��
=
�
????????????��
=
�
????????????��

EXERCISE 12.5
LAW OF COSINE:- USED WHEN 2 SIDES AND 1 ANGLE ARE GIVEN:
5. a
2
= b
2
+c
2
-2bc cosα
6. b
2
= c
2
+a
2
-2ca cosβ
7. c
2
= a
2
+b
2
-2ab cos�
8. Cosα =
�
2
+�
2
−�
2
2��

9. Cosβ =⁡
�
2
+�
2
−�
2
2��

10. Cos� =⁡
�
2
+�
2
−�
2
2��

LAW OF TANGENTS:- USED WHEN 2 SIDES & 2 ANGLES ARE GIVEN:
11.
�−�
�+�
=
���⁡
�−�
2
���
�+�
2

12.
�−�
�+�
=
���⁡
�−�
2
���
�+�
2

13.
�−�
�+�
=
���⁡
�−�
2
���
�+�
2

5. P a g e |5

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
EXERCISE 12.6
HALF ANGLE FORMULAS
NOTE:- IN ALL THESE HALF ANGLE FORMULAS: 2s = a+b+c
33.⁡�??????�
�
2
=⁡√
(�−�)(�−�)
��

34.⁡�??????�
�
2
=⁡√
(�−�)(�−�)
��

35.⁡�??????�
�
2
=⁡√
(�−�)(�−�)
��

36.⁡���
�
2
=⁡√
�(�−�)
��

37.⁡���
�
2
=⁡√
�(�−�)
��

38.⁡���
�
2
=⁡√
�(�−�)
��

39.⁡���
�
2
=⁡√
(�−�)(�−�)
�(�−�)

40.���
�
2
=⁡√
(�−�)(�−�)
�(�−�)

41.���
�
2
=⁡√
(�−�)(�−�)
�(�−�)

EXERCISE 12.7
TO FIND THE AREA OF TRIANGLES
Let area of the triangle is :-⁡??????
CASE 1:- IF TWO SIDES AND ONE ANGLE ARE GIVEN:
23:- Δ=⁡
1
2
��⁡�??????��=
1
2
��⁡�??????��=
1
2
��⁡�??????��
CASE 2:- IF ONE SIDE AND TWO ANGLES ARE GIVEN:
24:-⁡Δ=
�
2
⁡????????????��⁡????????????��
2????????????��

25:-⁡Δ=
�
2
⁡????????????��⁡????????????��
2????????????��

6. P a g e |6

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
26:-⁡Δ=
�
2
⁡????????????��⁡????????????��
2????????????��

CASE 3:- IF ONLY THREE SIDES ARE GIVEN:
27:- Δ=√�(�−�)(�−�)(�−�)⁡ .`. (Hero`s formula)
EXERCISE 12.8
28:- R =
���
4Δ

29:- r =
Δ
�

30:- r1 =
Δ
�−�

31:- r2 =
Δ
�−�

32:- r3=
Δ
�−�

HALF ANGLE FORMULAS
NOTE:- IN ALL THESE HALF ANGLE FORMULAS: 2s = a+b+c
33.⁡�??????�
�
2
=⁡√
(�−�)(�−�)
��

34.⁡�??????�
�
2
=⁡√
(�−�)(�−�)
��

35.⁡�??????�
�
2
=⁡√
(�−�)(�−�)
��

36.⁡���
�
2
=⁡√
�(�−�)
��

37.⁡���
�
2
=⁡√
�(�−�)
��

38.⁡���
�
2
=⁡√
�(�−�)
��

39.⁡���
�
2
=⁡√
(�−�)(�−�)
�(�−�)

40.���
�
2
=⁡√
(�−�)(�−�)
�(�−�)

7. P a g e |7

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
41.���
�
2
=⁡√
(�−�)(�−�)
�(�−�)


FORMULAS CH APTER:-13
CLASS:-XI
1. Sin
-1
A + Sin
-1
B = Sin
-1
(A√1−�
2
+ B√1−�
2
)
2. Sin
-1
A - Sin
-1
B = Sin
-1
(A√1−�
2
- B√1−�
2
)
3. Cos
-1
A +Cos
-1
B = Cos
-1
(AB-√1−�
2
√1−�
2
)
4. Cos
-1
A - Cos
-1
B = Cos
-1
(AB+√1−�
2
√1−�
2
)
5. Tan
-1
A + Tan
-1
B = Tan
−1
(
�+�
1−��
)
6. Tan
-1
A - Tan
-1
B = ⁡Tan
−1
(
�−�
1+��
)
7. 2Tan
-1
A – Tan
-1
B = Tan
-1
⁡(
2�
1−�
2
)

8. 2 Cos
-1
A = Cos
-1
(A
2
– 1)

8. P a g e |8

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
FORMULAS CH APTER:-6
CLASS:-XI
EXERCISE:- 6.2
1. an = a1 + (n-1)d .`. USED TO FIND N
TH
TERM OF THE ARITHMETIC
PROGRATION. [A.P]
EXERCISE:- 6.3
2. A.M =
�+�
2

3. an = a1 + (n-1)d
EXERCISE:- 6.4
4. Sn =
�
2
(a1 + an )
5. Sn =
�
2
[2a1 + (n-1)d ] .`. USED TO SUM THE TERMS OF THE ARITHMETIC SERIES.
EXERCISE:- 6.5
NOT FOR LAHORE BOARD.
EXERCISE:- 6.6
6. an = a1 r
n-1
.`. USED TO FIND N
TH
TERM OF THE GEOMETRIC
PROGRATION. [G.P]
EXERCISE:- 6.7
7. G.P = √�.� or, [A
1/2
. B
1/2
]
8. an = a1 r
n-1

EXERCISE:- 6.8
9. Sn =
�
1
1−�
.`. USED TO SUM THE TERMS OF THE GEOMETRIC SERIES TO
INFINITY.
10. Sn =
�
1
⁡[1⁡−⁡�
??????
]
1⁡−⁡�
.`. USED TO SUM THE TERMS OF THE GEOMETRIC SERIES TO
“n” TERM WHEN “r” IS LESS THEN “1”.

9. P a g e |9

MR.Munawer
CLASS:- XI
ST
-YEAR F Fazaia Inter college Lahore (session 2016-17)
11. Sn =
�
1
⁡[�
??????
−1]
�−1
.`. USED TO SUM THE TERMS OF THE GEOMETRIC SERIES TO
“n” TERM WHEN “r” IS GREATER THEN “1”.

EXERCISE:- 6.9
NOT FOR LAHORE BOARD.
EXERCISE:- 6.10
12. H.P =
2⁡�.�
�⁡+⁡�

13. an = a1 + (n-1)d
EXERCISE:- 6.11
14.⁡∑⁡1⁡
�
??????=1
= n .’. (n is put in the place of simple“1”)
15. ∑⁡k⁡
�
??????=1
= Tk =
�(�+1)
2

16. ∑⁡K
2�
??????=1
= Tk
2
=
�(�+1)(2�+1)
6

17. ∑⁡k
3
⁡
�
??????=1⁡= Tk
3
= [
�(�+1)
2
]
2


NOTE:- IN THESE FORMULAS,
n = n
th
term.
d = difference between two arithmetic terms.
R = ratio between two geometric terms.
Sn = Sum to “n” terms.
Tk = TOTAL “k”
IMPORTANT NOTES:-
G
2
= AH
A < G < H