Converting Radians to Degrees

Introduction

• Conversion of angles from radian measure to degree measure.
• Utilize a unit fraction based on the equivalence:
  • (2\pi) radians = 360 degrees.

Unit Fractions

• Deriving unit fractions:
  1. (\pi / 180^\circ = 1)
     • Derived from dividing (2\pi = 360^\circ) by (360^\circ).
  2. (180^\circ / \pi = 1)
     • Derived from dividing (2\pi = 360^\circ) by (2\pi).
• Usage:
  • Convert from radians to degrees using (180^\circ / \pi).
  • Convert from degrees to radians using (\pi / 180^\circ).

Conversion Examples

Example 1

• Convert to Degrees:
  • Multiply by (180^\circ / \pi).
  • Simplify (\pi) out from numerator and denominator.
  • Simplification of constants:
    • 6 and 180: Common factor of 6.
    • Calculation: 5 * 30 = 150 degrees.
  • Result: (150^\circ)

Example 2

• Convert to Degrees:
  • Simplify (\pi) out.
  • Simplification of constants:
    • 9 and 180: Common factor of 9.
    • Calculation: (-2 * 20 = -40^\circ).
  • Result: (-40^\circ)

Example 3

• Convert Decimal Radians to Degrees:
  • Multiply by (180^\circ / \pi).
  • When no (\pi) is present, nothing simplifies.
  • Use a calculator for approximation:
    • Calculation: 2.1 * 180 / \pi.
    • Approximate Result: (12.3^\circ)

Conclusion

• Key conversion method involves multiplying by the appropriate unit fraction.
• Simplification where possible is crucial for exact results.
• Use of calculators for decimal approximations when (\pi) is not in the radian measure.
• Understanding of these conversions is critical for solving angle problems in trigonometry.