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″