Concept of Angle



  • Part of Angle - Initial Side,Terminal Side,Vertex
  • Types of Angle - Positive and Negative Angles
  • Measuring Angles in Radian
  • Measuring Angles in Degrees
  • initial side, terminal side, vertex,positive angle, negative angle
  • Degree measure
  • Radian measure
  • Relation between radian and real numbers
  • Relation between degree and radian

Definition of Angle

What is an angle? In-Plane Geometry, a figure formed by two rays or lines that share a common endpoint is called an angle. The word “angle” is derived from the Latin word “angulus”, which means “corner”. The two rays are called the sides of an angle, and the common endpoint is called the vertex.

One complete revolution from the position of the initial side, as indicated in Fig


Positive Angle: If the angle goes counterclockwise, it is called a positive angle.

Negative Angle: If the angle goes clockwise, it is called a negative angle.

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side, and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is positive, and if the direction of rotation is clockwise, then the angle is negative.

Degree measure- If a rotation from the initial side to the terminal side is `(1/360)"th"` of a revolution, the angle is said to have a measure of one degree, written as 1°.  

A degree is divided into 60 minutes and a minute into 60 seconds. One-sixtieth of a degree is called a minute, written as 1′, and one-sixtieth of a minute is called a second, written as 1″.

Thus, 1° = 60′, 1′ = 60″ Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are shown in Fig 

Radian measure - There is another unit for the measurement of an angle called the radian measure. The angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian. 
 The figures show the angles whose measures are `1  radian, –1  radian, 1  1/2  radian and -1  1/2  radian.`

Thus, if in a circle of radius r, an arc of length l subtends an angle θ radian at the centre, we have
`θ= l/r` or `l= rθ.`
Here, θ will always be represented in terms of radian.
Relation between radian and real numbers:
Consider the unit circle with centre O. Let A be any point on the circle. Consider OA as the initial side of an angle. Then the length of an arc of the circle will give the radian measure of the angle at which the arc will subtend at the centre of the circle. Consider the line PAQ which is tangent to the circle at A. Let point A represent the real number zero, AP represents a positive real number, and AQ represents negative real numbers (Fig). Suppose we rope the line AP in the anticlockwise direction along the circle and AQ in the clockwise direction. In that case, every real number will correspond to a radian measure and conversely. Thus, radian measures and real numbers can be considered the same.

Relation between degree and radian: 

Circumference of a circle =2πr
Number of arcs= 2π
One full rotation= 2π Radians
2π Radians= 360°
π= 180°
`π/2` Radians= 90°
1 Radian= `"360°"/"2π"` ∼ 57.2958°

The relation between degree measures and radian measure of some common angles are given in the following table:

Notational Convention:
Since angles are measured either in degrees or in radians, we adopt the convention that whenever we write angle θ°, we mean the angle whose degree measure is θ and whenever we write angle β, we mean the angle whose radian measure is β. Note that the word ‘radian’ is frequently omitted when an angle is expressed in radians.

Thus, `π= 180°` and `π/4= 45°` are written with the understanding that `π` and `π/4` are radian measures. Thus, we can say that

`"Radian measure" = π/180xx "Degree measure"`

`"Degree measure" = 180/π xx "Radian measure"` 

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