Angular Measure Notes

Radian measure

  • Radian measure of a central angle in a circle: the angle in radians is the arc length divided by the radius.
  • Formula: \Theta = \frac{A}{R}
    where A is the length of the subtended arc and R is the radius.
  • Arc-length form: A = R\,\Theta
  • Example: A central angle in a circle of radius 3 ft subtends an arc of 6 in.
    • Convert radius to inches: R = 3\text{ ft} = 36\text{ in}
    • \Theta = \frac{A}{R} = \frac{6}{36} = \frac{1}{6}\text{ rad}
  • From now on, radian measure is the default angular measure.

Sector area and relationships

  • For a circle of radius R and central angle \Theta, the sector has area S and subtends arc length A.
  • Formulas:
    • Arc length: A = R\,\Theta
    • Sector area: S = \frac{R^{2}\,\Theta}{2}
  • Alternative derivation: S = \frac{\Theta}{2\pi} \cdot \pi R^{2} = \frac{R^{2}\,\Theta}{2}
  • Key idea: Given any two of the quantities (\Theta, R, A, S), the other two can be found.

Worked examples

  • Example 1: Given \Theta = 2 and R = 3\text{ in}, find A and S.
    • A = R\,\Theta = 3\cdot 2 = 6\text{ in}
    • S = \frac{R^{2}\,\Theta}{2} = \frac{3^{2}\cdot 2}{2} = 9\text{ in}^2
  • Example 2: Given S = 3\text{ in}^2 and A = 4\text{ in}, find R and \Theta.
    • From A = R\,\Theta\Rightarrow \Theta = \frac{A}{R} = \frac{4}{R}
    • From S = \frac{R^{2}\,\Theta}{2}: 3 = \frac{R^{2}\cdot \Theta}{2}
    • Substitute \Theta = \frac{4}{R}: 3 = \frac{R^{2}\cdot (4/R)}{2} = \frac{4R}{2} = 2R\Rightarrow R = \frac{3}{2}\text{ in}
    • Then \Theta = \frac{4}{R} = \frac{4}{3/2} = \frac{8}{3}\text{ rad}

Homework (Problems 1–4)

  • In problems 1–4, A = arc length (in inches) and S = sector area (in square inches) for a circle of radius R\text{ in} subtended by central angle \Theta\text{ rad}.
    1) Given \Theta = \frac{\pi}{8}, R = 2, find A and S.
    2) Given R = 3, S = 10, find \Theta and A.
    3) Given \Theta = \frac{\pi}{12}, A = 2\pi, find R and S.
    4) Given S = 3\pi, A = 2\pi, find R and \Theta.

Degrees, minutes, seconds

  • Key relationships:
    • 2\pi = 360^{\circ}
    • 1\ \text{rad} = \dfrac{180^{\circ}}{\pi}
    • 1^{\circ} = \dfrac{\pi}{180} \ \text{rad}
    • 1^{\circ} = 60\'; 1\' = 60\"; 1^{\circ} = 60\'\,; 1^{\circ} = 3600\"
    • (Thus: 1^{\circ} = 60\' = \tfrac{1}{60}^{\circ}\,; 1^{\prime} = 60\".)

Examples: radian–degree conversions

  • Convert (6^{\circ} 16\' 36\") to radians:
    • Result: \Theta = \dfrac{1883\,\pi}{54000}
  • Convert (\dfrac{\pi}{7}) radian to degrees, minutes, seconds:
    • Calculation: \dfrac{180^{\circ}}{7} = 25^{\circ} + \dfrac{42\'}{ } + \dfrac{51.428\"}{ }
    • Approximate result: 25^{\circ}\,42\'\,51.43\"

Homework (cont’d)

  • 5) Convert (1^{\circ}\,12\'\,54\") to radians
  • 6) Convert (3^{\circ}\,5\'\,24\") to radians
  • 7) Convert (\dfrac{\pi}{81}) radian to D°M′S″
  • 8) Convert (\dfrac{\pi}{96}) radian to D°M′S″