Binomial Theorem

  • ( 1+x )n = 1+nx+[n( n-1)/2!] .x2 + [n(n-1)(n-2)/3!].x3 +……
  • ( 1+x )-n = 1-nx+[-n( n+1)/2!] .x2 – [n(n+1)(n+2)/3!].x3 +……

If x<<1 then x2,x3,…. is negligible. so:

  • (1+x ) -n ≈ 1-nx
  • (1-x ) n ≈ 1-nx
  • (1-x ) -n ≈ 1+nx

Exponential Series

  • ex = 1 + x/1! + x2 /2! + x3/3! + …..
  • e = 1 + 1/1! + 1/2! + 1/3! + ….
  • e = 2.7182
  • e-x = 1 – x/1! + x2 /2! – x3/3! + …..
  • ex + e-x = 2 [ 1 + x2/2! + x4/4 + …..]

Logrithm formula

Formulae of Logarithm

  • loga mn = loga m + loga n
  • loga m/n = loga m – loga n
  • loga mn = n loga m
  • loga m= logb m × loga b
  • loge m = 2.3026 log10 m
  • log10 m = 0.4343 loge m

Logrithmic Series

  • loge ( 1+x ) = x – x2/2 + x3/3 – x4/4 +……
  • loge ( 1-x ) = -[x + x2/2 + x3/3 + x4/4 +……]
  • loge ( 1+x ) / ( 1-x ) = 2 [x + x3/3 + x5/5 +……]

Factors

  • ( a+b )2 = a2 + b2 + 2ab
  • ( a-b )2 = a2 + b2 – 2ab
  • ( a2 – b2) = ( a+b ) ( a-b )
  • ( a2 + b2 ) = ( a+b )2– 2ab
  • ( a+b )3 = a3 + b3 + 3ab( a+b )
  • ( a-b )3 = a3 – b3 – 3ab( a-b )
  • ( a+b+c)2 = a2 + b2 + c2 +2(ab + bc + ac )
  • a3 + b3 + c3 – 3 abc = ( a+b+c ) ( a2 + b2 + c2 – ab – bc – ac )
  • ( a+b )4 = a4 + b4 + 2ab ( 2a2 + 3ab + 2b2)
  • ( a-b )4 = a4 + b4 – 2ab ( 2a2 + 3ab – 2b2 )

Differentiation formula

Fomulae of Differentiation

  • ddx (c) = 0   Where c is constant.
  • ddx (cx) = c   Where c is constant.
  • dudt = dudxdxdt
  • ddx (u+v) = dudx + dvdx
  • ddx (uv) = u dvdx + v dudx
  • ddx ( xn ) = n xn-1  Where n is real number.
  • ddx un = n un-1  dudx   Where u is function of x.
  • ddx sin x = cos x
  • ddx cos x = -sin x
  • ddx tan x = sec2 x
  • ddx cot x = – cosec2 x
  • ddx sec x = tan x sec x
  • ddx cosec x = – cot x cosec x
  • ddx loge x = 1x
  • ddx loge u = 1u   dudx
  • ddx ( ex ) = ex
  • ddx ( eax ) = a eax
  • ddx sin ax = a cos ax
  • ddx sin (ax+b) = a cos (ax+b)
  • ddx cos ax = – a sin ax
  • ddx cos (ax+b) = – a sin (ax+b)

Integration formula

Integration formula

  • ∫ xn dx = xn+1n+1   where ( n≠ -1 )
  • ∫ dx = x
  • ∫ c xn dx = c xn+1n+1   where ( n≠ -1 )
  • 1x dx = ln x
  • ∫ sin x dx = – cos x
  • ∫ cos x dx = sin x
  • ∫ ex dx = ex
  • ∫ ( u+v ) dx = ∫ u dx + ∫ v dx
  • ∫ sec2 x dx = tan x
  • ∫ cosec2 x dx = – cot x
  • ∫ sec x tan x dx = sec x
  • ∫ cosec x cot x dx = – cosec x
  • ∫ eax dx = 1a eax
  • ∫ sin ax dx = – 1a cos ax
  • ∫ sin (ax+b) dx = – 1a cos (ax+b)
  • ∫ cos ax dx = 1a sin ax
  • ∫ cos (ax+b) dx = 1a sin (ax+b)

Trigonometry formula

Basic trigonometry formula

  • Perp./Hyp. = sin θ
  • Base/Hyp. = cos θ
  • Perp./Base = tan θ
  • Base/Perp. = cot θ
  • Hyp./Base = sec θ
  • Hyp./Perp. = cosec θ

Baudhayana formula

( Hyp. )2 = ( Base )2 + ( Perp.)2

Some common trigonometric Formulas

  • sin(A+B) = sin A cos B + cos A sin B
  • sin(A-B) = sin A cos B – cos A sin B
  • cos(A+B) = cos A cos B – sin A sin B
  • cos(A-B) = cos A cos B + sin A sin B
  • sin 2A = 2 sin A cos B
  • cos 2A = cos2 A – sin2 A
  • cos 2A = 1 – 2 sin2 A
  • sin(A+B) + sin(A-B) = 2 sin A cos B
  • sin(A+B) – sin(A-B) = 2 cos A sin B
  • cos(A+B) + cos(A-B) = 2 cos A cos B
  • cos(A+B) – cos(A-B) = -2 sin A sin B

Trigonometric Ratios

Trigonometry ratio table:

Table of trigonometrical ratios of some standard angels:

Angle sin θ cos θ tan θ
00 0 1 0
300 $$\frac{1}{2}$$ $$\frac{\sqrt{3}}{2}$$ $$\frac{1}{\sqrt{3}}$$
450 $$\frac{1}{\sqrt{2}}$$ $$\frac{1}{\sqrt{2}}$$ 1
600 $$\frac{\sqrt{3}}{2}$$ $$\frac{1}{2}$$ $$\sqrt{3}$$
900 1 0 $$\infty $$
1200 $$\frac{\sqrt{3}}{2}$$ $$-\frac{1}{2}$$ $$-\sqrt{3}$$
1350 $$\frac{1}{\sqrt{2}}$$ $$-\frac{1}{\sqrt{2}}$$ -1
1500 $$\frac{1}{2}$$ $$-\frac{\sqrt{3}}{2}$$ $$-\frac{1}{\sqrt{3}}$$
1800 0 -1 0
2700 -1 0 $$-\infty $$
3600 0 1 0
Angle cot θ sec θ cosec θ
00 $$\infty $$ 1 $$\infty $$
300 $$\sqrt{3}$$ $$\frac{2}{\sqrt{3}}$$ 2
450 1 $$\sqrt{2}$$ $$\sqrt{2}$$
600 $$\frac{1}{\sqrt{3}}$$ 2 $$\frac{2}{\sqrt{3}}$$
900 0 $$\infty $$ 1
1200 $$-\frac{1}{\sqrt{3}}$$ -2 $$\frac{2}{\sqrt{3}}$$
1350 -1 $$-\sqrt{2}$$ $$\sqrt{2}$$
1500 $$-\sqrt{3}$$ $$-\frac{2}{\sqrt{3}}$$ 2
1800 $$-\infty $$ -1 $$\infty $$
2700 -1 0 $$\infty $$
3600 $$\infty $$ 1 $$\infty $$

Relation between Trigonometric Ratios

  • sin θ cosec θ = 1
  • cos θ sec θ = 1
  • tan θ cot θ = 1
  • tan θ = sin θ/cos θ
  • cot θ = cos θ/sin θ
  • sin2 θ + cos2 θ = 1
  • 1 + tan2 θ = sec2 θ
  • 1 + cot2 θ = cosec2 θ

A. Trigonometric Ratios of acute angles

  • Perp./Hyp. = sin θ
  • Base/Hyp. = cos θ
  • Perp./Base = tan θ
  • Base/Perp. = cot θ
  • Hyp./Base = sec θ
  • Hyp./Perp. = cosec θ

B. Trigonometric ratios of allied angles


1. Trigonometric ratios of (-θ) in terms of (θ)

sin(-θ) = -sinθ
cos(-θ) = cosθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
sec(-θ) = secθ
cosec(-θ) = -cosecθ


2. Trigonometric ratios of (900-θ) in terms of (θ)

sin(900-θ) = cosθ
cos(900-θ) = sinθ
tan(900-θ) = cotθ
cot(900-θ) = tanθ
sec(900-θ) = cosecθ
cosec(900-θ) = secθ


3. Trigonometric ratios of (900+θ) in terms of (θ)

sin(900+θ) = cosθ
cos(900+θ) = -sinθ
tan(900+θ) = -cotθ
cot(900+θ) = -tanθ
sec(900+θ) = -cosecθ
cosec(900+θ) = secθ


4. Trigonometric ratios of (1800-θ) in terms of (θ)

sin(1800-θ) = sinθ
cos(1890-θ) = -cosθ
tan(1800-θ) = -tanθ
cot(1800-θ) = -cotθ
sec(1800-θ) = -secθ
cosec(1800-θ) = cosecθ


5. Trigonometric ratios of (1800+θ) in terms of (θ)

sin(1800+θ) = -sinθ
cos(1800+θ) = -cosθ
tan(1800+θ) = tanθ
cot(1800+θ) = cotθ
sec(1800+θ) = -secθ
cosec(1800+θ) = -cosecθ


6. Trigonometric ratios of (900+θ) in terms of (θ)

sin(2700-θ) = -cosθ
cos(2700-θ) = -sinθ
tan(2700-θ) = cotθ
cot(2700-θ) = tanθ
sec(2700-θ) = -cosecθ
cosec(2700θ) = -secθ


7. Trigonometric ratios of (900+θ) in terms of (θ)

sin(2700+θ) = -cosθ
cos(2700+θ) = sinθ
tan(2700+θ) = -cotθ
cot(2700+θ) = -tanθ
sec(2700+θ) = cosecθ
cosec(2700+θ) = -secθ


8. Trigonometric ratios of (3600-θ) in terms of (θ)

sin(3600-θ) = -sinθ
cos(3600-θ) = cosθ
tan(3600-θ) = -tanθ
cot(3600-θ) = -cotθ
sec(3600-θ) = secθ
cosec(3600-θ) = -cosecθ


9. Trigonometric ratios of (3600-θ) in terms of (θ)

sin(3600+θ) = sinθ
cos(3600+θ) = cosθ
tan(3600+θ) = tanθ
cot(3600+θ) = cotθ
sec(3600+θ) = secθ
cosec(3600+θ) = cosecθ


10. Trigonometric ratios of (n×3600±θ) in terms of (θ)

sin(n×3600±θ) = ±sinθ
cos(n×3600±θ) = cosθ
tan(n×3600±θ) = ±tanθ
cot(n×3600±θ) = ±cotθ
sec(n×3600±θ) = secθ
cosec(n×3600±θ) = ±cosecθ

C. Trigonometric ratios of compound angels


1. Trigonometric ratios of sum and difference of two angles

  • sin(A+B) = sinA cosB + cosA sinB
  • cos(A+B) = cosA cosB – sinA sinB
  • sin(A-B) = sinA cosB – cosA sinB
  • cos(A-B) = cosA cosB + sinA sinB


2. Transformation of product into sums of differences

  • 2 sinA cosB = sin(A+B) + sin(A-B)
  • 2 cosA sinB = sin(A+B) – sin(A-B)
  • 2 cosA cosB = cos(A+B) + cos(A-B)
  • 2 sinA sinB = cos(A+B) – cos(A-B)


3. Transformation of sum or difference into product

Suppose A+B=C and A-B=D
or $$ A = \frac{C+D}{2}$$ and $$B = \frac{C-D}{2}$$

  • $$sinC+sinD = 2 sin \frac{C+D}{2} cos\frac{C-D}{2}$$
  • $$sinC-sinD = 2 cos \frac{C+D}{2} sin\frac{C-D}{2}$$
  • $$cosC+cosD = 2 cos \frac{C+D}{2} cos\frac{C-D}{2}$$
  • $$cosC-cosD = 2 sin\frac{C+D}{2} sin\frac{D-C}{2}$$


4. Trigonometric ratios of sum of more than two angles

  • sin(A+B+C) = sinA cosB cos C + cosA sinB cosC + cosA cosB sinC – sinA sinB sinC
  • cos(A+B+C) = cosA cosB cosC – sinA sinB cosC – sinA cosB sinC – cosA sinB sinC

D. Trigonometric ratios of multiple and sub-multiple angles

Multiple angles: 2A, 3A, 4A ……

Sub-multiple angles : $$ \frac{A}{2}, \frac{A}{3}, \frac{A}{4}$$…….

1. Trigonometric ratios of an angle 2A in terms of angle A

  • sin2A = 2sinA cosA
  • cos2A = 1-2sin2A
  • $$ \ tan2A = \frac{2tanA}{1-tan^2A}$$

2. Trigonometric ratios of sin2A and cos2A in terms of tanA

  • $$ \ sin2A = \frac{2tanA}{1+tan^2A}$$
  • $$ \ cos2A = \frac{1-tan^2A}{1+tan^2A}$$

3. Trigonometric ratios of an angle 3A in terms of angle A

  • sin3A = 3 sinA – 4sin3A
  • cos3A = 4 cos3A – 3 cosA
  • $$ \ tan3A = \frac{3 tanA – tan^3A}{1-3 tan^2A}$$

4. Trigonometric ratios of an angle 180

  • $$sin18^0 = \frac{-1+\sqrt{5}}{4}$$
  • $$cos18^0 = \frac{\sqrt{10+2\sqrt{5}}}{4}$$

5. Trigonometric ratios of an angle 360

  • $$ cos36^0 = \frac{1+\sqrt{5}}{4}$$
  • $$ sin36^0 = \frac{\sqrt{10-2\sqrt{5}}}{4}$$

6. Trigonometric ratios of an angle A in terms of angle A/2.

  • $$ sinA = 2 sin\frac{A}{2}cos\frac{A}{2}$$
  • $$ cosA = 1- 2 sin^2\frac{A}{2}$$
  • $$ tanA = \frac{2tan\frac{A}{2}}{1-tan^2\frac{A}{2}}$$
  • $$ sinA = \frac{2tan\frac{A}{2}}{1+tan^2\frac{A}{2}}$$
  • $$ cosA = \frac{1-tan^2\frac{A}{2}}{1+tan^2\frac{A}{2}}$$

7. Trigonometric ratios of an angle \(\frac{A}{2}\) in terms of cosA

  • $$sin\frac{A}{2} = \pm \sqrt{\frac{1-cosA}{2}}$$
  • $$cos\frac{A}{2} = \pm \sqrt{\frac{1+cosA}{2}}$$
  • $$tan\frac{A}{2} = \pm \sqrt{\frac{1-cosA}{1+cosA}}$$

8. Trigonometric ratios of an angle \(\frac{A}{2}\) in terms of sinA

  • $$sin\frac{A}{2} + cos\frac{A}{2} = \pm \sqrt{1+sinA}$$
  • $$sin\frac{A}{2} – cos\frac{A}{2} = \pm \sqrt{1-sinA}$$

Series Expnsion of Trigonometric functions

  • sin θ = θ – θ3/3! + θ5/5! – θ7/7! …..
  • cos θ = 1 – θ2/2! + θ4/4! – θ6/6! …..
  • tan θ = θ + θ3/3 + 2θ5/15 …..

Approximate Value

If is θ small

  • sin θ ≈ θ
  • cos θ ≈ 1
  • tan θ ≈ θ

Average Value

  • < sin θ > = < sin nθ > = 0
  • < cos θ > = < cos nθ > = 0
  • < sin2 θ > = < sin2 nθ > = 1/2
  • < cos2 θ > = < cos2 nθ > = 1/2