Circular Measure: Radians and Arc Length

Introduction to Radian Measure

• Definition of Radian: One radian is the measure of an angle subtended at the centre of a circle by an arc whose length is the same as the radius of the circle ($s = r$).
• Notation: Radians are denoted as "rad", $1^c$, or $1^r$.
• Fundamental Relationship: A complete rotation of $360^\circ$ is equal to $2\pi\,rad$. Therefore, $\pi\,rad = 180^\circ$.

Conversion Formulas

• Degrees to Radians: Multiply the angle in degrees by $\frac{\pi}{180^\circ}$.
• Radians to Degrees: Multiply the angle in radians by $\frac{180^\circ}{\pi}$.
• Constant Value: In calculations, use $\pi = 3.142$ unless otherwise specified.

Determining Arc Length

• Arc Length Formula: The length of an arc, $s$, for a circle with radius $r$ and an angle $\theta$ measured in radians is given by:
  • $s = r\theta$
• Proportionality: Arc length is proportional to the angle subtended at the centre: $\frac{\text{Minor arc length AB}}{2\pi r} = \frac{\theta}{2\pi}$.

Perimeter of a Circle Segment

• Definition: The perimeter of a segment is the sum of the arc length and the length of the chord.
• Chord Length Calculation: The chord length can be found using:
  • Cosine Rule: $AC^2 = OA^2 + OC^2 - 2(OA)(OC)\cos(\angle AOC)$.
  • Trigonometric Ratios: By drawing a perpendicular from the centre to the chord, length $= 2r\sin\left(\frac{\theta}{2}\right)$.

Mathematical Applications

• Circular Measure Utility: These formulas are applied to real-world objects like bicycle tires (radius $31\,cm$), shot put regions, running tracks, and cross-sections of train tunnels.
• Historical Note: Gottfried Wilhelm Leibniz introduced a method to calculate $\pi$ without a circle using the formula: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \dots$

Questions & Discussion

• Question: 1 radian is smaller than $60^\circ$. What are the advantages of using angles in radians compared to angles in degrees? Discuss.
• Question: From the definition of radian, can you derive the formula $s = r\theta$?
• Question: Can the length of AC be obtained using sine rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$?