Overview
This lecture covers fundamental concepts about angles in the unit circle, their measurement, and methods for converting between decimal degrees and degrees-minutes-seconds.
Angles and Their Representation
- An angle is formed by two non-collinear rays sharing a common endpoint (vertex).
- The initial side is where the angle starts; the terminal side is where the rotation ends.
- The amount of rotation from the initial to the terminal side defines the angle.
- Counterclockwise rotation produces positive angles; clockwise rotation gives negative angles.
- A full rotation equals 360 degrees; negative angles denote clockwise direction.
Units of Angle Measurement
- Angles can be measured in degrees (°) or radians (not covered in detail here).
- One degree equals 60 minutes (1° = 60′); one minute equals 60 seconds (1′ = 60″).
- The degree system is based on the sexagesimal (base-60) numeral system developed by the Babylonians.
Converting Decimal Degrees to Degrees, Minutes, Seconds (DMS)
- Separate the whole number (degrees) from the decimal part.
- Multiply the decimal part by 60 to get minutes.
- If the minutes part has a decimal, multiply it by 60 to get seconds.
- Example: 22.4° = 22° 24′; 14.21° = 14° 12′ 36″.
Converting Degrees, Minutes, Seconds (DMS) to Decimal Degrees
- Keep degrees as is.
- Convert minutes to degrees by dividing by 60.
- Convert seconds to degrees by dividing by 3600.
- Add all to get decimal degrees.
- Example: 31° 12′ 54″ = 31 + 12/60 + 54/3600 = 31.215°.
Key Terms & Definitions
- Angle — Formed by two rays with a common endpoint (vertex).
- Initial Side — The starting position of the angle.
- Terminal Side — The ending position after rotation.
- Degree (°) — A unit of angle measure; 1 full circle = 360°.
- Minute (′) — 1/60th of a degree.
- Second (″) — 1/60th of a minute or 1/3600th of a degree.
- Positive Angle — Angle formed by counterclockwise rotation.
- Negative Angle — Angle formed by clockwise rotation.
- Decimal Degree — Expression of an angle in degrees with a decimal fraction.
Action Items / Next Steps
- Practice converting between decimal degrees and DMS using a calculator.
- Review the concepts of positive and negative angle measurements.
- Prepare for exercises on angle conversions for the next lesson.