### 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 .....

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

### How many fundamental and supplementary units in SI?

In SI, there are seven base ( fundamental ) units and two supplementary units.
Base unit

Physical quantity

Unit

Symbol

Mass

Kilogram

kg

Length

Metre

m

Time

Second

s

Temperature

Kelvin

K

Electric current

Ampere

A

Luminous intensity

Candela

cd

Quantity of matter

Mole

mol

Supplementary units

Physical quantity

Unit

Symbol

Plane angle

Solid angle

sr

Conversion factor
Conversion of length

1 centimetre = 10-2 metre
1 millimetre = 10-3metre
1 micrometre = 10-6metre
1 nanometre = 10-9 metre
1 angstrom= 10-10 metre
1 fermi = 10-15 metre
1 kilometre = 103 metre
1 austronomical unit = 1AU=1.496 × 1011 metre
1 light year = 1 ly = 9.461 ×1015metre
1 mile = 1.609 ×103 metre
1 yard = 0.9144 metre
1 inch = 0.0254 metre

Conversion of time

1 mili second= 10-3 second
1 micro second = 10-6 second
1 neno second = 10-9 second
1 hour = 60 minute = 3600 second
1 day = 24 hours =86400 second
1 year = 365 day = 3.156× 107 second
1 sec = 10-8second

Conversion of mass

1 gram = 10 -3 kg
1 quintal = 100 kg
1 tonne = 1000 kg
1 slug = 14.59…

### What is the Universal law of gravitation?

Force of attraction between two masses $$m_1$$ and $$m_2$$ is given by, $$F = \frac{m_1 m_2}{r^2}$$ Where, r is the distance between two masses $$m_1$$ and $$m_2$$. G is a constant, called the Universal gravitational constant. That is called Universal law of gravitation. Acceleration due to gravity ( variation formula ) If a body is falling freely, under the effect of gravity, then the acceleration in the body is called acceleration due to gravity. Variation in acceleration due to gravity with height Acceleration due to gravity at a height h above the surface of earth is, $$g' = g (1 - \frac{2h}{R_e})$$ Where, Re is the radius of the earth. g is the acceleratin due to gravity. Variation in acceleration due to gravity with depth Acceleration due to gravity at a depth d below the surface of earth is, $$g' = g (1 - \frac{d}{R_e})$$ Where, Re is the radius of the earth. g is the acceleratin due to gravity. Variation in acceleration due to gravity with rotatio…

### Differentiation formula

Fomulae of Differentiation d⁄dx (c) = 0   Where c is constant.d⁄dx (cx) = c   Where c is constant.du⁄dt = du⁄dx ⋅ dx⁄dtd⁄dx (u+v) = du⁄dx + dv⁄dxd⁄dx (uv) = u dv⁄dx + v du⁄dxd⁄dx ( xn ) = n xn-1  Where n is real number.d⁄dx un = n un-1du⁄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 loge x = 1⁄xd⁄dx loge u = 1⁄udu⁄dxd⁄dx ( ex ) = exd⁄dx ( eax ) = a eaxd⁄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)