This document contains formulas and concepts related to trigonometry, coordinate geometry, and sequences and series from classes XI. It includes over 50 formulas for trigonometric identities, sum and difference identities, half-angle formulas, laws of sines, cosines and tangents, area of triangles, and inverse trigonometric functions. It also includes formulas for the nth term and sum of terms for arithmetic and geometric progressions. The document is a reference sheet for students to use when working through exercises in their math textbook.

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