127-6.1

Angles, Arc Length, and Circular Motion

Measuring Angles

  • Degrees: Common unit for measuring angles.

    • One full revolution is equivalent to 360 degrees.

    • Origin of 360 possibly from Babylonian calendar (360 days in a year).

  • Radians: Another unit of measure for angles.

    • One full revolution equals 2π radians.

    • Conversion relations:

      • ( 2π = 360^{\circ} )

      • ( π = 180^{\circ} )

Conversion Between Degrees and Radians

  • From degrees to radians: Multiply degrees by ( \frac{π}{180} )

  • From radians to degrees: Multiply radians by ( \frac{180}{π} )

  • Examples:

    • Convert ( 60^{\circ} ) to radians:

      • Calculation: ( 60 \times \frac{π}{180} = \frac{1}{3}π )

    • Convert ( -405^{\circ} ) to radians:

      • Calculation: ( -405 \times \frac{π}{180} = -\frac{9}{4}π )

    • Convert ( \frac{3}{4}π ) to degrees:

      • Calculation: ( \frac{3}{4}π \times \frac{180}{π} = 135^{\circ} )

  • Convert ( -\frac{2}{3}π ) to degrees:

    • Calculation: ( -\frac{2}{3}π \times \frac{180}{π} = -120^{\circ} )

Degrees, Minutes, Seconds (DMS)

  • A degree is divided into 60 minutes, each minute into 60 seconds.

Conversion Formulae

  • The following conversions hold:

    • 1 minute (') = ( \frac{1}{60} ) degree

    • 1 second (") = ( \frac{1}{3600} ) degree

Example Conversion to DMS

  • Write 83 degrees, 24 minutes, 13 seconds as decimal degrees:

    • Calculation: ( 83 + \frac{24}{60} + \frac{13}{3600} = 83.4036^{\circ} )

  • Convert ( 163.36^{\circ} ) to DMS:

    • Split: ( 163 + 0.36 )

    • Calculation: ( 0.36 \times 60 = 21.6, )

    • Then, ( 0.6 \times 60 = 36,

    • Therefore: ( 163.36^{\circ} = 163^{\circ} 21' 36'' )

Angles in Standard Position

  • Definition: An angle is in standard position if its vertex is at the origin and initial side is on the positive x-axis.

  • Positive Angles: Measured counterclockwise.

  • Negative Angles: Measured clockwise.

Drawing and Coterminal Angles

  • Quadrants: Identify which quadrant the angle lies in based on its measurement.

    • Example: Draw ( \frac{6}{5}π ): Convert to degrees first: ( 6 \times 180/π = 216^{\circ} ), which lies in Quadrant II.

    • For ( -135^{\circ} ): Lies in Quadrant III as it is measured clockwise.

  • Coterminal Angles: Angles that share the same terminal side, derived by adding or subtracting full revolutions (360° or 2π radians).

Arc Length and Sector Area

Arc Length Formula

  • Formula for Arc Length:

    • ( S = rθ ) (arc length, radius, angle in radians)

  • Example: Find the arc length of a circle with radius 3 inches and angle ( \frac{3}{4}π ):

    • Calculation:

      • ( S = 3 \times \frac{3}{4}π = \frac{9}{4}π ) inches.

Area of a Sector Formula

  • Formula for Area of a Sector:

    • ( A = \frac{1}{2}r^2θ ) (area, radius, angle in radians)

  • Example: Area of a sector with radius 4 inches and angle ( \frac{3}{4}π ):

    • Calculation:

      • ( A = \frac{1}{2} \times 4^2 \times \frac{3}{4}π = 12π ) square inches.

Angular and Linear Speed

Angular Speed Formula

  • Formula: ( ω = \frac{θ}{t} ) (angle in radians over time)

  • Example: If a gear rotates at 75 rpm:

    • Convert to radians per minute: ( 75 \times 2π = 150π , \text{rad/min} )

Linear Speed Formula

  • Formula: ( v = rω ) (radius multiplied by angular speed)

  • Example: For a point 3mm from the center with angular speed 150π:

    • Calculation:

      • ( v = 3 \times 150π = 450π , \text{mm/min} )

  • Further example for a wheel rotating at 2160 rpm and calculating its linear speed at 30 cm radius also demonstrate this relation.