Angles, Arcs, and Their Measures: Foundations of Trigonometry

Chapter 9: The Unit Circle and the Functions of Trigonometry

Section 9.1: Angles, Arcs, and Their Measures

This section introduces the fundamental terminology and concepts related to angles, their measurement, and their classification, which are essential for understanding trigonometric functions and the unit circle.

Basic Terminology

• Line AB: Determined by two distinct points A and B, it extends infinitely in both directions.

• Line Segment AB: The portion of the line that includes points A and B and all points between them.

• Ray AB: The part of the line that starts at A and passes through B, extending infinitely in one direction.

• Angle: Formed by rotating a ray (the initial side) about its endpoint (the vertex) to a new position (the terminal side).

Positive angles are generated by counterclockwise rotation; negative angles by clockwise rotation.

Degree Measure

• The degree is a unit for measuring angles, dividing a circle into 360 equal parts.

• There are 360° in one full rotation.

• Acute angle: Between 0° and 90°.

• Right angle: Exactly 90°.

• Obtuse angle: Greater than 90° but less than 180°.

• Straight angle: Exactly 180°.

Complementary and Supplementary Angles

• Complementary angles: Two positive angles whose sum is 90°.

• Supplementary angles: Two positive angles whose sum is 180°.

Example: If two angles are (6m)° and (3m)°, and their sum is 90°, then 6m + 3m = 90°, so m = 10°, and the angles are 60° and 30°.

Calculating with Degrees, Minutes, and Seconds

• One minute (1') is $\frac{1}{60}$ of a degree. Thus, $60' = 1°$.

• One second (1") is $\frac{1}{60}$ of a minute, or $\frac{1}{3600}$ of a degree. Thus, $60'' = 1'$ and $3600'' = 1°$.

Example: To add $51°29'$ and $32°46'$, add degrees and minutes separately, converting excess minutes to degrees as needed.

Converting Between Decimal Degrees and Degrees, Minutes, and Seconds

• To convert from degrees, minutes, and seconds to decimal degrees:

  • Decimal degrees = degrees + (minutes/60) + (seconds/3600)

• To convert from decimal degrees to degrees, minutes, and seconds:

  • Degrees = integer part

  • Minutes = integer part of (decimal part × 60)

  • Seconds = (remaining decimal part × 60)

Example: Convert $74°8'14''$ to decimal degrees: $74 + \frac{8}{60} + \frac{14}{3600} = 74.1372°$ (rounded).

Example: Convert $34.817°$ to degrees, minutes, and seconds:

• Degrees: 34

• Minutes: $0.817 \times 60 = 49.02 \approx 49$

• Seconds: $0.02 \times 60 = 1.2 \approx 1$

• So, $34.817° = 34°49'1''$

Standard Position Angles

• Angles in standard position have their vertex at the origin and their initial side along the positive x-axis.

• Quadrantal angles are in standard position with terminal sides along the x- or y-axis (e.g., 0°, 90°, 180°, 270°, 360°).

• Coterminal angles share the same initial and terminal sides but may have different measures (e.g., 30° and 390°).

Example: 420° and 60° are coterminal because $420° - 360° = 60°$.

Additional info: These foundational concepts are essential for further study in trigonometry, including the unit circle, trigonometric functions, and their applications.