Category: Mathematics

Formulae of Logarithm

• log_a mn = log_a m + log_a n
• log_a m/n = log_a m – log_a n
• log_a mⁿ = n log_a m
• log_a m= log_b m × log_a b
• log_e m = 2.3026 log₁₀ m
• log₁₀ m = 0.4343 log_e m

Logrithmic Series

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

Fomulae of Differentiation

• ^d⁄_dx (c) = 0   Where c is constant.
• ^d⁄_dx (cx) = c   Where c is constant.
• ^du⁄_dt = ^du⁄_dx ⋅ ^dx⁄_dt
• ^d⁄_dx (u+v) = ^du⁄_dx + ^dv⁄_dx
• ^d⁄_dx (uv) = u ^dv⁄_dx + v ^du⁄_dx
• ^d⁄_dx ( xⁿ ) = n x^n-1  Where n is real number.
• ^d⁄_dx uⁿ = n u^{n-1  du}⁄_dx   Where u is function of x.
• ^d⁄_dx sin x = cos x
• ^d⁄_dx cos x = -sin x
• ^d⁄_dx tan x = sec² x
• ^d⁄_dx cot x = – cosec² x
• ^d⁄_dx sec x = tan x sec x
• ^d⁄_dx cosec x = – cot x cosec x
• ^d⁄_dx log_e x = ¹⁄_x
• ^d⁄_dx log_e u = ¹⁄_u   ^du⁄_dx
• ^d⁄_dx ( e^x ) = e^x
• ^d⁄_dx ( e^ax ) = a e^ax
• ^d⁄_dx sin ax = a cos ax
• ^d⁄_dx sin (ax+b) = a cos (ax+b)
• ^d⁄_dx cos ax = – a sin ax
• ^d⁄_dx cos (ax+b) = – a sin (ax+b)

Integration formula

• ∫ xⁿ dx = ^xn+1⁄_n+1   where ( n≠ -1 )
• ∫ dx = x
• ∫ c xⁿ dx = c ^xn+1⁄_n+1   where ( n≠ -1 )
• ∫ ¹⁄_x dx = ln x
• ∫ sin x dx = – cos x
• ∫ cos x dx = sin x
• ∫ e^x dx = e^x
• ∫ ( 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
• ∫ e^ax dx = ¹⁄_a e^ax
• ∫ sin ax dx = – ¹⁄_a cos ax
• ∫ sin (ax+b) dx = – ¹⁄_a cos (ax+b)
• ∫ cos ax dx = ¹⁄_a sin ax
• ∫ cos (ax+b) dx = ¹⁄_a sin (ax+b)

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