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)

**(c) Circular 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 1 degree (or 1°) and 1 minute ( or 1’ ) |
= 90 degrees (or 90°) = 60 minutes ( or 60’) = 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

1 right angle 1 grade ( or 1 ^{g})and 1 minute (or 1‵) |
= 100 grades (or, 100^{g})= 100 minutes (or, 100‵) = 100 seconds ( or, 100‶). |

**Note:** (i) Clearly, minute and second in sexagesimal and centesimal systems are different.

For example,

1 right angle = 90 × 60 = 5400 sexagesimal minutes = (5400)’

and 1 right angle = 100 × 100 = 10000 centesimàl minutes = (10000)‶

Therefore, 90° = 100

or, 1° = (10/9)

The first relation is used to reduce an angle of sexagesimal system to centesimal system and the second is used to reduce an angle of centesimal system to sexagesimal system.

**(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:

Circular (radian) measure of an angle:

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:

## Degrees0° 30° 45° 60° 90° 120° 135° 150° 180° 270° 360° |
## Radians0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π 3π/2 2π |

**●** **Measurement of Angles**

**Sign of Angles****Trigonometric Angles****Measure of Angles in Trigonometry****Systems of Measuring Angles****Important Properties on Circle****S is Equal to R Theta****Sexagesimal, Centesimal and Circular Systems****Convert the Systems of Measuring Angles****Convert Circular Measure****Convert into Radian****Problems Based on Systems of Measuring Angles****Length of an Arc****Problems based on S R Theta Formula**

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