Different Systems of Measurement of Angles in Trigonometry:
Sexagesimal System, Centesimal System and Circular System are 3 different systems of Measuring Angle in Trigonometry:
Angles in geometry are measured in terms of a right angle. The Right Angle serves as the foundation for defining the various systems for measuring angles. However, due to its size, this is an inconvenient unit of measurement. Generally, 3 different systems are used to measure angles. Following are the 3 different systems of measuring angle:
(1) Sexagesimal System ( or English System or British System)
(2) Centesimal System ( or French System)
(3) Circular System (Radian Measure)
(1) Sexagesimal System (Degree Measure):
An angle is measured in Degrees, Minutes and Seconds in the Sexagesimal system.
The Sexagesimal system is the most popular and widely used measurement system to measure angles. In this system, a Right Angle is divided into 90 equal parts known as Degrees. Also in this system, each degree is subdivided into 60 equal parts called Minutes, and each minute is subdivided into 60 equal parts known as Seconds.
A degree is denoted by the symbol 1°, a minute is denoted by the symbol 1', and a second is denoted by the symbol 1". As a result, we get
1 Right Angle = 90 Degrees (90°)
1 Degree (1°) = 60 Minutes (60')
1 Minute (1') = 60 Seconds (60")
Sexagesimal System is a well-established system that is always used in practical trigonometry applications. It is, however, not very convenient due to the multipliers 60 and 90.
(2) Centesimal System (Grade Measure):
An angle is measured in Grades, Minutes and Seconds in the Centesimal system.
The Centesimal System is another measurement system to measure angles in trigonometry. In this system, a Right Angle is divided into 100 equal parts known as Grades. Also in this system, each grade is subdivided into 100 equal parts called Minutes, and each minute is subdivided into 100 equal parts known as Seconds.
A grade is denoted by the symbol 1g, a minute is denoted by the symbol 1', and a second is denoted by the symbol 1". As a result, we get
1 Right Angle = 100 Grades (100g)
1 Grade (1g) = 100 Minutes (100')
1 Minute (1') = 100 Seconds (100")
Centesimal System would be far more user-friendly and convenient than the standard Sexagesimal System. However, before it could be implemented, a large number of tables would have to be recalculated. For this reason, the system has never been used in practice.
Important Remark:
1. It is obvious that minutes and seconds differ in sexagesimal and centesimal systems.
2. To convert Sexagesimal into Centesimal Measure: If the angle does not have an integral number of degrees, it can be reduced to a fraction of a degree before being converted to grades.
(3) Circular System (Radian Measure):
We all know angles can be measured in degrees. For example:
A straight angle = 180°
A right angle = 90°
A full circle = 360°
But this degree system was made by humans (historically 360 parts). In higher mathematics and science, we use a more natural system that comes directly from the circle itself. This is called the Circular System or the Radian Measure.
An angle is measured in Radian in the Circular system. Circular System is the third measurement system of measuring angles that have been devised, and it is used in all higher branches of mathematics and in other applications of Science.
What is a Radian? (The Unit of Circular System)
Imagine a circle with center O and radius r.
- Cut an arc (a small curved part of the circle).
- If the arc length equals the radius (that is, s = r), the angle that arc makes at the center is 1 radian.
Definition: 1 radian is the angle made at the center of a circle by an arc whose length equals the radius of the circle.
Core idea (ratio): angle in radians = arc length ÷ radius
╬╕ = s / r
Why do we say this is natural?
Because the definition of a radian does not depend on human-made numbers (like 360° or 400 grads). It comes directly from the geometry of the circle:
angle (in radians) = arc length / radius
How Many Radians in One Circle?
The circumference of a circle is 2╧Аr. Using ╬╕ = s / r:
╬╕ = (2╧Аr) / r = 2╧А
Therefore:
one complete revolution = 2╧А radians.
An angle is measured in Radian in the Circular system.
The unit used in the Circular System is obtained as follows:
| Angle | Degrees (°) | Radians |
|---|---|---|
| Full circle | 360° | 2╧А |
| Straight angle | 180° | ╧А |
| Right angle | 90° | ╧А/2 |
| One radian (approx.) | ≈ 57.2958° | 1 |
Degree–Radian Conversion (Quick Rules)
- 360° = 2╧А radians
- 180° = ╧А radians
- 1° = ╧А/180 radians
- 1 radian = 180/╧А degrees (≈ 57° 17′ 45″)
Why is the Radian So Important?
- Simple formulas
Arc length:s = r╬╕| Sector area:A = (1/2) r2 ╬╕(valid when╬╕is in radians). - Natural in calculus
Key limits hold in radians, e.g.,lim (╬╕ → 0) [sin ╬╕ / ╬╕] = 1. - Standard in physics
Waves, oscillations, circular motion: using radians keeps equations clean and dimensionless.
Think of it like this: Degrees are a human-made ruler for angles. Radians are the circle’s own ruler. That’s why advanced math and science use radians.
Tiny Memory Trick
Half circle is 180°. Half of 2╧А is ╧А. So remember:
180° ↔ ╧А radians.
From that, everything else falls into place.
Quick Examples
- Convert 60° to radians: 60 × (╧А/180) = ╧А/3
- Convert ╧А/6 radians to degrees:
(╧А/6) × (180/╧А) = 30° - Arc length with ╬╕ in radians: If
r = 5and╬╕ = ╧А/4, thens = r╬╕ = 5 × (╧А/4) = 5╧А/4.
In one line: Circular system measures angles in radians; 1 radian is the angle made by an arc equal to the radius, and a full circle is 2╧А radians.
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