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**Basic part **

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*Mathematical tools everyone should know*

*Mathematical tools everyone should know*

__ANGLE__: Amount of revolution of a given line about a fixed line is called the

Fig.1: Angle and Quadrants |

angle traced by the line.

●Negative, when taken in the clockwise direction.

(For angle rotation everyone should have an idea of quadrant.)

● Angle can be measured in degrees or radians.

Sexagimal System (degree)

Right angle = 90 °

1° =60' (60 minutes)

1'=60'' (60 seconds)

1' is the 60th division if 1°and 1" is the 60th division of x'. It should not be confused with minutes and seconds of a clock.

**CIRCULAR SYSTEM**

One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle as in fig.2.

Hence, length of the arc’s’ is related to angle in radians subtended at the center of the circle by this arc as

Fig.2. |

S = r

**θ**(θ called theta)
● Since, angle subtended by a complete circle at the center in radians will be

**θ**= S / r = 2πr / r (angle subtended by a complete circle).

**θ**= 2π

Therefore 360° = 2π radians

1 radian= 360°/2π = 180°/ π = 57.27°

Conversion of some angles from degrees into radians

1 radian =180°/π

1. 30° degree

1 radian = 180°/π

Degree = radian (just for students understanding)

30 = 180°/π

30 × π / 180= π/6

= 2. 45°

**Do yourself for exercise**60°,90°,120°, 135°, 180°, 210°,300°, 360°

Conversion of some angles from radians into degrees:

1. 2π/3 = 2/3×180°=120°

2. π/4 = 1/4× 180° = 45°

**Trigonometry Ratios:**

Perpendicular side always be opposite to angle

**θ**. And base is always opposite to right angle as in fig.4.**Value of T-ratio at Standard angle.**

fig. Values of T-ratio |

Allied angle is any
angle which can be written as 90° × n ±

**θ**where 'n = 0, 1, 2,.....'
1. Sin(

To Be Conti..... in part - 2

References: NCERT books, Moderns pub., Internet Sources

*A*+*B)*= sin*A c*os*B*+ sin*B*cos*A*

*2. S*in(*A -- B )*= sin*A*cos*B*-- sin*B*cos*A*

*3. C*os (A +*B)*= cos*A*cos*B*- sin*A*sin*B*

*4. C*os(A -*B)*= cos*A*cos*B*+ sin*A*sin*B***C - D Formulae**

1. Sin C + Sin D = 2 Sin[(C+D)/2]Cos[(C-D)/2]

2. SinC - SinD = 2 Cos[(C+D)/2]Sin[(C-D)/2]

3. CosC + CosD = 2 Cos[(C+D)/2]Cos[(C-D)/2]

4. CosC - CosD = 2 Sin[(C+D)/2]Sin[(D-C)/2]

for e.g

1. Sin 75° = ?

ans. Sin 75° = Sin (45° + 30°)

using sin(

*A*+*B)*= sin*A c*os*B*+ sin*B*cos*A**=*Sin 45° cos 30° + cos 45° sin 30°

using Value of T-Ratio

= (1/√2 ) (√3/2) + (1/√2) (1/2)

= [(√3 + 1 ) / (2√2)]

**Trigonometry ratio of multiple angle & Identity**

1. Cos2θ = Cos²θ - Sin²θ

= 2Cos²θ - 1

= 1 - 2Sin²θ

= (1-tan²θ)/(1+tan²θ)

= 2Cos²θ - 1

= 1 - 2Sin²θ

= (1-tan²θ)/(1+tan²θ)

2. Sin2θ = 2SinθCosθ = (2tanθ)/(1+tan²θ)

3. Sin²θ + Cos²θ = 1

4. Tan²θ + 1 = Sec²θ

5. Cot²θ + 1 = Cosec²θ

To Be Conti..... in part - 2

References: NCERT books, Moderns pub., Internet Sources

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