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1. Angle Basics
Angle: A rotation from an initial side to a terminal side about a vertex. Counterclockwise = positive, clockwise = negative.
Angle Types
| Type | Measure |
|---|
| Acute | 0° < θ < 90° |
| Right | θ = 90° |
| Obtuse | 90° < θ < 180° |
| Straight | θ = 180° |
| Reflex | 180° < θ < 360° |
Standard Position
Vertex at origin, initial side on positive x-axis. Terminal side determines the quadrant.
Coterminal Angles
θ ± 360°n (degrees) or θ ± 2πn (radians)
Complementary and Supplementary
Complement: 90° − θ Supplement: 180° − θ
2. Radian Measure
Radian: The angle whose intercepted arc equals the radius. One full rotation = 2π radians.
180° = π radians
Conversion Formulas
Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π
Common Conversions Table
| Degrees | Radians | Degrees | Radians |
|---|
| 0° | 0 | 210° | 7π/6 |
| 30° | π/6 | 225° | 5π/4 |
| 45° | π/4 | 240° | 4π/3 |
| 60° | π/3 | 270° | 3π/2 |
| 90° | π/2 | 300° | 5π/3 |
| 120° | 2π/3 | 315° | 7π/4 |
| 135° | 3π/4 | 330° | 11π/6 |
| 150° | 5π/6 | 360° | 2π |
| 180° | π | | |
3. Arc Length and Sector Area
Arc Length: s = rθ (θ in radians)
Sector Area: A = (1/2)r²θ (θ in radians)
Always convert degrees to radians before using these formulas!
4. Angular and Linear Speed
Angular Speed: ω = θ/t (rad per unit time)
Linear Speed: v = s/t = rω
Key relationship: v = rω
Connected gears/pulleys: r1ω1 = r2ω2
5. Study Tips
- Memorize the degree-radian conversion table. It will be used throughout the course.
- Always check: is the angle in radians before using s = rθ or A = (1/2)r²θ?
- Practice converting between rpm and rad/s fluently.
- Draw pictures for every problem involving angles in standard position.