Chapter 1: Circular Measure

1.1 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. (p.3)
• 2π rad = 360°; π rad = 180°. (p.3)
• 1 rad = 180°/π ≈ 57.29°; 1° = π/180 ≈ 0.01746 rad. (p.3)
• Radians to degrees: multiply by 180°/π. (p.3)
• Degrees to radians: multiply by π/180°. (p.4)
• Special angles: 30° = π/6, 36° = π/5, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. (p.4)

1.2 Arc Length of a Circle

• Arc length is proportional to the angle at the centre: minor arc AB = (2πr/360°) × θ (θ in degrees). (p.5)
• Arc length, s = rθ, where θ is in radians. (p.6)
• A reflex angle satisfies 180° < θ < 360°; convert it to radians before using s = rθ. (p.6)
• Perimeter of a segment = arc length + chord length. (p.8)
• Chord length can be found with the cosine rule, AC² = r² + r² − 2r² cos θ. (p.8)
• Alternative for the chord: AC = 2r sin(θ/2), using the perpendicular from the centre to the chord. (p.8)
• Problems on arc length: add the arcs and straight sides that enclose the region. (p.9)

1.3 Area of Sector of a Circle

• Area of minor sector AOB = (πr²/360°) × θ (θ in degrees). (p.12)
• Area of sector, A = ½r²θ, where θ is in radians. (p.13)
• Since s = rθ, area of sector A = ½rs. (p.13)
• If θ is in degrees, A = (θ/360°) × πr². (p.18)
• Area of segment = area of sector − area of triangle. (p.15)
• Area of triangle = ½ab sin C (or ½ac sin B, ½bc sin A). (p.16)
• Heron's formula: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2. (p.16)
• Area of a region between two sectors with the same centre = larger sector − smaller sector. (p.17)

1.4 Application of Circular Measures

• Reflex angle at the centre = 2π − θ (in radians). (p.21)
• Area of a major segment = area of triangle + area of major sector. (p.21)
• Volume of a prism-like slice = cross-section area × height. (p.22)