PURE MATHEMATICS - Trigonometry

 

Radians

 

 

Radians

Arc Length

Sector Area

Small Angles

 

 

 

What is a 'Radian'?

 

 

radian definition

 

 

A radian is the angle subtended at the centre of a circle by an arc the same length as the radius of the circle.

 

Units

1C (meaning 1 radian)= 57.296 deg.

 

 

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Arc Length

 

 

radian definition #2

 

 

The arc length is proportional to its subtended angle.

 

Hence, if θ(theta) is in degrees and 'l' is the arc length:

 

 

equation for arc length

 

 

An angle can be expressed in radians by dividing the arc length by the radius.

 

Therefore θ in radians is given by:

 

 

arc length

 

 

Therefore for a circle(a 360 deg. angle), where the arc length is '2πr' (two pi r)and the radius is 'r' , the number of radians is 2πr/r , i.e. 2π .

 

 

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Sector Area

 

The area of a sector is proportional to the angle its arc subtends at the centre.

 

If a sector contains an angle of θo then its area is given by:

 

 

sector area

 

 

However, if θ is in radians, remembering there are 2π radians in a circle:

 

 

sector area #2

 

 

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Small Angles

 

 

small angles

 

 

For small angles(<10 deg.) there is a convergence between the value of the angle in radians with the value of its sine & tangent.

 

This approximate sine value may be expressed as:

 

 

sine theta approximates t o theta

 

 

The approximate cosine value is obtained thus:

 

 

small angle - cosine equation

 

 

 

 

 

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