Systems of Measuring Angles

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The following three different systems of units are used in the measurement of trigonometrical angles :

(a) Sexagesimal System (or English System)

(b) Centesimal System (or French System)

If a straight line stands on another line and if the two adjacent angles thus formed are equal to one another, then by geometry, each of these angles is called a right angle. This right angle forms the basis in defining the different systems for the measurement of angles.

Definition of Systems of Measuring Angles:

(a) Sexagesimal System:

In Sexagesimal System, an angle is measured in degrees, minutes and seconds. A complete rotation describes 360°. In this system, a right angle is divided into 90 equal parts and each such part is called a Degree (1°); a degree is divided into 60 equal parts and each such part is called a Sexagesimal Minute (1’) and a minute is further sub-divided into 60 equal parts, each of which is called a Sexagesimal Second (1’’). In short,

1 right angle = 90 degrees (or 90°)

1 degree (or 1°) = 60 minutes (or 60’)

1 minute (or 1’) = 60 seconds (or 60’’)

(b) Centesimal System:

In Centesimal System, an angle is measured in grades, minutes and seconds. In this system, a right angle is divided into 100 equal parts and each such part is called a Grade (1^g); again, a grade is divided into 100 equal parts and each such part is called a Centesimal Minute (1‵); and a minute is further sub-divided into 100 equal parts, each of which is called a Centesimal Second (1‶). In short,

1 right angle = 100 grades (or, 100^g)

1 grade (or 1^g ) = 100 minutes (or, 100‵)

1 minute (or 1‵) = 100 seconds (or, 100‶).

Note:

Clearly, minute and second in sexagesimal and centesimal systems are different. For example,

Since, 1 right angle = 90° = 100^g

Therefore, 90° = 100^gor, 1° = (10/9)^g and 1^g= (9/10)°

(c) Circular System:

In this System, an angle is measured in radians. In higher mathematics angles are usually measured in circular system. In this system a radian is considered as the unit for the measurement of angles.

Definition of Radian: A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius.

A radian defined as follows:

In any circle, the angle subtended at its centre by an arc of the circle whose length is equal to the radius of the circle is called a radian. Let OX = r be the radius of a circle having center at O.

Now, take an arc XY of the circle such that arc XY = r and join OY. By definition, ∠XOY = one radian.

One radian is written as 1^c, 2 radians as 2^c and in general, k radians as k^c.

Circular (radian) measure of an angle:

Degrees	Radians
0° 30° 45° 60° 90° 120° 135° 150° 180° 270° 360°	0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π 3π/2 2π

The circular measure of an angle is the number of radians it contains.

Thus the circular (radian) measure of a right angle is π/2.

If an angle is given without mentioning units, it is assumed to be in radians. The relation between degree measures and circular (radian) measures of some standard angles are given below: