1. Systems of Angle Measurement

An angle is the figure formed by two rays sharing a common endpoint. Depending on the system used, the measurement of a single right angle varies.

System Base Unit 1 Right Angle Sub-units
Sexagesimal (English) Degree (°) 90° 1° = 60', 1' = 60"
Centesimal (French) Grade (g) 100g 1g = 100m, 1m = 100s
Circular Radian (c) \(\frac{\pi}{2}\)c Defined by arc/radius

2. Conversion Relationships

To convert between systems, we use the fundamental constant that all these systems represent the same rotation. If \(D\), \(G\), and \(R\) are the measures of the same angle in Degrees, Grades, and Radians respectively:

\[\frac{D}{90} = \frac{G}{100} = \frac{2R}{\pi}\]

Useful Derived Formulas:

3. Relation Between Arc, Radius, and Angle

In a circle with radius \(r\), an arc of length \(l\) subtends a central angle \(\theta\). The relationship is given by:

\[\theta = \frac{l}{r}\]

CRITICAL RULE: In this formula, the angle \(\theta\) MUST be expressed in radians. If the angle is given in degrees or grades, you must convert it to radians first before applying the formula.

4. Worked Examples

Example 1: Mixed System Conversion

Convert 75° into Grades and Radians.

(a) To Grades:
Using \(\frac{D}{90} = \frac{G}{100}\)
\(\frac{75}{90} = \frac{G}{100} \Rightarrow \frac{5}{6} = \frac{G}{100}\)
\(G = \frac{500}{6} = 83.33^g\)

(b) To Radians:
Using \(\frac{D}{180} = \frac{R}{\pi}\)
\(\frac{75}{180} = \frac{R}{\pi} \Rightarrow \frac{5}{12} = \frac{R}{\pi}\)
\(R = \frac{5\pi}{12}^c\)

Example 2: Arc Length Application

Find the angle in degrees subtended by an arc of length 11cm in a circle of radius 14cm.

Solution:
Given: \(l = 11\text{cm}\), \(r = 14\text{cm}\)
\(\theta = \frac{l}{r} = \frac{11}{14}\text{ radians}\)

Now, convert radians to degrees using \(D = R \times \frac{180}{\pi}\):
\(D = \frac{11}{14} \times \frac{180}{22/7}\)
\(D = \frac{11}{14} \times \frac{180 \times 7}{22}\)
\(D = 45^\circ\)