Formulas , formulae for class xi math

1. P a g e |1 NAME:-____________________ CLASS/SEC:- __________
MR.Munawer
CLASS:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
FORMULAS CHAPTER:-10
CLASS:-XI
EXERCISE:-10.1
1:- α+β+ 𝛾 =1800
2:- 900
-α =β
3:- 900
-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 𝛼+tan⁡𝛽
1−tan 𝛼⁡tan⁡𝛽
9:- Tan(α-β)=⁡
tan 𝛼−tan⁡𝛽
1+tan 𝛼⁡tan⁡𝛽
10:- Cot(α+β)=⁡
cot 𝛼 cot 𝛽−1
cot 𝛼+cot⁡𝛽
11:- Cot(α-β)=⁡
cot 𝛼 cot 𝛽+1
cot 𝛼−cot⁡𝛽
EXERCISE:-10.3

2. P a g e |2
MR.Munawer
CLASS:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
12:- Sin2α =2SinαCosα
13:- Cos2α = Cos2
α-Sin2
α or :- 2cos2
α -1 or :- 1-2sin2
α
14:- Tan2α =⁡
2tan⁡α
1−tan2 𝛼
15:- Sin
α
2
= √
1−𝑐𝑜𝑠𝛼
2
16:- Cos
α
2
= √
1+𝑐𝑜𝑠𝛼
2
17:- Tan
α
2
=√
1−𝑐𝑜𝑠𝛼
1+𝑐𝑜𝑠𝛼
18:- Sin3α = 3Sinα-4Sin3
α
19:- Cos3α = 4Cos3
α-3Cosα
20:- Tan3α =⁡
3𝑡𝑎𝑛𝛼−tan3 𝛼
1−3 tan2 𝛼
21:-Cos2Ø =⁡
1−tan2 Ø
1+tan2 Ø
22:-Sin2Ø =⁡
2 𝑡𝑎𝑛 Ø
1+tan2 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(
𝑃+𝑄
2
) Cos(
𝑃−𝑄
2
)
28:- Sin(P)-Sin(Q) =2Cos(
𝑃+𝑄
2
) Sin(
𝑃−𝑄
2
)
29:-Cos(P)+Cos(Q) =2Cos(
𝑃+𝑄
2
) Cos(
𝑃−𝑄
2
)
30:- Cos(P)-Cos(Q) = -2Sin(
𝑃+𝑄
2
) Sin(
𝑃−𝑄
2
)

3. P a g e |3
MR.Munawer
CLASS:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
FORMULAS CHAPTER:-11
CLASS:-XI
-:FOR DOMAIN AND RANGE OF TRIGNOMETRICFUNCTIONS:-
-:PERIODS OF TRIGONOMETRIC FUNCTIONS:-
1:- 2πis the period of Cosθ.
2:- 2πis the period of Sinθ.
3:- 2πis the period of Cosecθ.
4:- 2πis the period of Secθ.
5:- πis the period of Tanθ.
6:- πis the period of Cotθ.
NOTE: [ πIS THE ONLY PERIODS OF Tanθ AND Cotθ. WHILE 2πIS THE PERIODS
OF ALL REMAINING TRIGONOMETRIC FUNCTIONS.]

4. P a g e |4
MR.Munawer
CLASS:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
FORMULAS CHAPTER:-12
CLASS:-XI
EXERCISE 12.4
LAW OF SINES:- USED WHEN 2 ANGLES & ONE SIDE OR 2 SIDES & 1 ANGLE ARE GIVEN:
1.
𝑎
𝑆𝑖𝑛𝛼
=
𝑏
𝑆𝑖𝑛𝛽
2.⁡
𝑏
𝑆𝑖𝑛𝛽
=
𝑐
𝑆𝑖𝑛𝛾
3.⁡
𝑎
𝑆𝑖𝑛𝛼
=
𝑐
𝑆𝑖𝑛𝛾
4.
𝑎
𝑆𝑖𝑛𝛼
=
𝑏
𝑆𝑖𝑛𝛽
=
𝑐
𝑆𝑖𝑛𝛾
EXERCISE 12.5
LAW OF COSINE:- USED WHEN 2 SIDES AND 1 ANGLE ARE GIVEN:
5. a2
= b2
+c2
-2bc cosα
6. b2
= c2
+a2
-2ca cosβ
7. c2
= a2
+b2
-2ab cos 𝛾
8. Cosα =
𝑏2+𝑐2−𝑎2
2𝑏𝑐
9. Cosβ =⁡
𝑐2+𝑎2−𝑏2
2𝑐𝑎
10. Cos 𝛾 =⁡
𝑎2+𝑏2
−𝑐2
2𝑎𝑏
LAW OF TANGENTS:- USED WHEN 2 SIDES & 2 ANGLES ARE GIVEN:
11.
𝑎−𝑏
𝑎+𝑏
=
𝑡𝑎𝑛⁡
𝛼−𝛽
2
𝑡𝑎𝑛
𝛼+𝛽
2
12.
𝑏−𝑐
𝑏+𝑐
=
𝑡𝑎𝑛⁡
𝛽−𝛾
2
𝑡𝑎𝑛
𝛽+𝛾
2
13.
𝑐−𝑎
𝑐+𝑎
=
𝑡𝑎𝑛⁡
𝛾−𝛼
2
𝑡𝑎𝑛
𝛾+𝛼
2

5. P a g e |5
MR.Munawer
CLASS:- XIST-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.⁡𝑆𝑖𝑛
𝛼
2
=⁡√
(𝑠−𝑏)(𝑠−𝑐)
𝑏𝑐
34.⁡𝑆𝑖𝑛
𝛽
2
=⁡√
(𝑠−𝑐)(𝑠−𝑎)
𝑎𝑐
35.⁡𝑆𝑖𝑛
𝛾
2
=⁡√
(𝑠−𝑎)(𝑠−𝑏)
𝑎𝑏
36.⁡𝐶𝑜𝑠
𝛼
2
=⁡√
𝑠(𝑠−𝑎)
𝑏𝑐
37.⁡𝐶𝑜𝑠
𝛽
2
=⁡√
𝑠(𝑠−𝑏)
𝑐𝑎
38.⁡𝐶𝑜𝑠
𝛾
2
=⁡√
𝑠(𝑠−𝑐)
𝑎𝑏
39.⁡𝑇𝑎𝑛
𝛼
2
=⁡√
(𝑠−𝑏)(𝑠−𝑐)
𝑠(𝑠−𝑎)
40.𝑇𝑎𝑛
𝛽
2
=⁡√
(𝑠−𝑐)(𝑠−𝑎)
𝑠(𝑠−𝑏)
41.𝑇𝑎𝑛
𝛾
2
=⁡√
(𝑠−𝑎)(𝑠−𝑏)
𝑠(𝑠−𝑐)
EXERCISE 12.7
TO FIND THE AREA OF TRIANGLES
Let area of the triangle is :-⁡𝚫
CASE 1:- IF TWO SIDES AND ONE ANGLE ARE GIVEN:
23:- Δ =⁡
1
2
𝑏𝑐⁡𝑆𝑖𝑛𝛼=
1
2
𝑐𝑎⁡𝑆𝑖𝑛𝛽=
1
2
𝑎𝑏⁡𝑆𝑖𝑛𝛾
CASE 2:- IF ONE SIDE AND TWO ANGLES ARE GIVEN:
24:-⁡Δ =
𝑎2⁡𝑆𝑖𝑛𝛽⁡𝑆𝑖𝑛𝛾
2𝑆𝑖𝑛𝛼
25:-⁡Δ =
𝑏2⁡𝑆𝑖𝑛𝛼⁡𝑆𝑖𝑛𝛾
2𝑆𝑖𝑛𝛽

6. P a g e |6
MR.Munawer
CLASS:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
26:-⁡Δ =
𝑐2⁡𝑆𝑖𝑛𝛼⁡𝑆𝑖𝑛𝛽
2𝑆𝑖𝑛𝛾
CASE 3:- IF ONLY THREE SIDES ARE GIVEN:
27:- Δ = √ 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)⁡ .`. (Hero`s formula)
EXERCISE 12.8
28:- R =
𝑎𝑏𝑐
4Δ
29:- r =
Δ
𝑠
30:- r1 =
Δ
𝑠−𝑎
31:- r2 =
Δ
𝑠−𝑏
32:- r3=
Δ
𝑠−𝑐
HALF ANGLE FORMULAS
NOTE:- IN ALL THESE HALF ANGLE FORMULAS: 2s = a+b+c
33.⁡𝑆𝑖𝑛
𝛼
2
=⁡√
(𝑠−𝑏)(𝑠−𝑐)
𝑏𝑐
34.⁡𝑆𝑖𝑛
𝛽
2
=⁡√
(𝑠−𝑐)(𝑠−𝑎)
𝑎𝑐
35.⁡𝑆𝑖𝑛
𝛾
2
=⁡√
(𝑠−𝑎)(𝑠−𝑏)
𝑎𝑏
36.⁡𝐶𝑜𝑠
𝛼
2
=⁡√
𝑠(𝑠−𝑎)
𝑏𝑐
37.⁡𝐶𝑜𝑠
𝛽
2
=⁡√
𝑠(𝑠−𝑏)
𝑐𝑎
38.⁡𝐶𝑜𝑠
𝛾
2
=⁡√
𝑠(𝑠−𝑐)
𝑎𝑏
39.⁡𝑇𝑎𝑛
𝛼
2
=⁡√
(𝑠−𝑏)(𝑠−𝑐)
𝑠(𝑠−𝑎)
40. 𝑇𝑎𝑛
𝛽
2
=⁡√
(𝑠−𝑐)(𝑠−𝑎)
𝑠(𝑠−𝑏)

7. P a g e |7
MR.Munawer
CLASS:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
41. 𝑇𝑎𝑛
𝛾
2
=⁡√
(𝑠−𝑎)(𝑠−𝑏)
𝑠(𝑠−𝑐)
FORMULAS CHAPTER:-13
CLASS:-XI
1. Sin-1
A + Sin-1
B = Sin-1
(A√1 − 𝐵2 + B√1 − 𝐴2)
2. Sin-1
A - Sin-1
B = Sin-1
(A√1 − 𝐵2 - B√1 − 𝐴2)
3. Cos-1
A +Cos-1
B = Cos-1
(AB-√1 − 𝐴2 √1 − 𝐵2)
4. Cos-1
A - Cos-1
B = Cos-1
(AB+√1 − 𝐴2 √1 − 𝐵2)
5. Tan-1
A + Tan-1
B = Tan−1
(
𝐴+𝐵
1−𝐴𝐵
)
6. Tan-1
A - Tan-1
B = ⁡Tan−1
(
𝐴−𝐵
1+𝐴𝐵
)
7. 2Tan-1
A – Tan-1
B = Tan-1
⁡(
2𝐴
1−𝐴2)
8. 2 Cos-1
A = Cos-1
(A2
– 1)

8. P a g e |8
MR.Munawer
CLASS:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
FORMULAS CHAPTER:-6
CLASS:-XI
EXERCISE:- 6.2
1. an = a1 + (n-1)d .`. USED TO FIND NTH
TERM OF THE ARITHMETIC
PROGRATION. [A.P]
EXERCISE:- 6.3
2. A.M =
𝐴+𝐵
2
3. an = a1 + (n-1)d
EXERCISE:- 6.4
4. Sn =
𝑛
2
(a1 + an )
5. Sn =
𝑛
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 rn-1
.`. USED TO FIND NTH
TERM OF THE GEOMETRIC
PROGRATION. [G.P]
EXERCISE:- 6.7
7. G.P = √ 𝐴. 𝐵 or, [A1/2
. B1/2
]
8. an = a1 rn-1
EXERCISE:- 6.8
9. Sn =
𝑎1
1−𝑟
.`. USED TO SUM THE TERMS OF THE GEOMETRIC SERIES TO
INFINITY.
10. Sn =
𝑎1⁡[1⁡−⁡𝑟 𝑛]
1⁡−⁡𝑟
.`. 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:- XIST-YEAR F Fazaia Inter college Lahore (session 2016-17)
11. Sn =
𝑎1⁡[𝑟 𝑛−1]
𝑟−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⁡𝐴.𝐵
𝐴⁡+⁡𝐵
13. an = a1 + (n-1)d
EXERCISE:- 6.11
14.⁡∑ ⁡1⁡𝑛
𝑘=1 = n .’. (n is put in the place of simple“1”)
15. ∑ ⁡k⁡𝑛
𝑘=1 = Tk =
𝑛(𝑛+1)
2
16. ∑ ⁡K2𝑛
𝑘=1 = Tk2
=
𝑛(𝑛+1)(2𝑛+1)
6
17. ∑ ⁡k3
⁡𝑛
𝑘=1 ⁡= Tk3
= [
𝑛( 𝑛+1)
2
]
2
NOTE:- IN THESE FORMULAS,
n = nth
term.
d = difference between two arithmetic terms.
R = ratio between two geometric terms.
Sn = Sum to “n” terms.
Tk = TOTAL “k”
IMPORTANT NOTES:-
G2
= AH
A < G < H