Math127 Angles, Arc Length, and Circular Motion

Angle Measurement

• There are different ways to measure angles:

  • Degrees: One complete circle equals 360 degrees.

  • Radians: One complete circle equals 2π radians.

Important Angle Conversions

• Examples of common angle measurements:

  • 0 degrees = 0 radians

  • 90 degrees = π/2 radians

  • 180 degrees = π radians

  • 270 degrees = 3π/2 radians

Converting Between Degrees and Radians

• Degrees to Radians: Multiply degrees by π/180.

• Radians to Degrees: Multiply radians by 180/π.

• Note that positive angles are measured counterclockwise, while negative angles are measured clockwise.

Arc Length and Central Angle

• Angles are represented by θ in formulas related to circles.

• When calculating arc lengths or area of sectors, θ must be in radians.

Examples of Angle Conversion

• Convert 45 degrees to radians:

  • Calculation: 45 * π/180 = π/4 radians

• Convert 150 degrees to radians:

  • Calculation: 150 * π/180 = 5π/6 radians

• Convert -90 degrees to radians:

  • Calculation: -90 * π/180 = -π/2 radians

• Convert π/9 to degrees:

  • Calculation: π/9 * 180/π = 20 degrees

• Convert 11π/6 to degrees:

  • Calculation: 11π/6 * 180/π = 330 degrees

• Convert 3 radians to degrees:

  • Calculation: 3 * 180/π ≈ 171.89 degrees

Drawing Angles in Standard Form

• Standard forms of angles help identify which quadrant they lie in.

• Example angles in standard form:

  • π/5: Located before π in the first quadrant.

  • 8π/3: Convert to mixed number (2 and 2π/3) indicating it’s in the second quadrant after one full rotation.

Coterminal Angles

• Definition: Coterminal angles share the same terminal side.

• Example: -120 degrees's coterminal angle within 360 degrees is:

  • Calculation: 360 - 120 = 240 degrees.

• Further coterminal angle could be: -120 - 360 = -480 degrees.

Example Analysis of Mixed Numbers

• For angles like 17π/6:

  • Convert to mixed number (2 and 5π/6).

  • Determine quadrant based on 5π/6; since it is before π, it is in the second quadrant.

Finding Arc Length

• Formula for arc length: Arc Length = Radius × θ

• Example 1: Calculate arc length with radius 3 inches and θ = π/3:

  • Arc Length = 3 × π/3 = π inches.

• Example 2: Calculate arc length with radius 7 inches and θ = 45 degrees:

  • Convert 45 degrees to radians (π/4).

  • Arc Length = 7 × π/4 = 7π/4 inches.

Area of the Sector

• Formula for the area of a sector: Area = 1/2 × Radius² × θ

• Example: Radius = 2 and θ = 30 degrees:

  • Convert 30 degrees: 30 * π/180 = π/6.

  • Area = 1/2 × 2² × π/6 = 2π/6 = π/3.

Rounding and Final Results

• Calculate area using rounding rules, e.g., round π/3 to two decimal places: 1.047/3.14 ≈ 1.05 square feet.