Angles and Their Measures 4.1

Overview

This lecture covers the basics of measuring angles, including degree and radian measure, conversions, special angle types, coterminal angles, and applications such as arc length, linear and angular speed, and area of a sector.

Angles and Their Measure

• An angle is formed by rotating a ray from its initial side to its terminal side, with the vertex as the rotation point.
• An angle in standard position has its vertex at the origin and initial side along the positive x-axis.
• Angles are positive if rotated counterclockwise, negative if clockwise.
• A full rotation is 360 degrees; one degree is 1/360 of a rotation.

Sketching and Classifying Angles

• Positive angles rotate counterclockwise from the x-axis; negative angles clockwise.
• 120° puts the terminal side in quadrant II; -70° in quadrant IV.
• Right angle = 90° (quarter turn); straight angle = 180° (half turn).
• Acute angle < 90°; obtuse angle is between 90° and 180°.

Complementary and Supplementary Angles

• Complementary: Two angles sum to 90° (e.g., 30° + 60°).
• Supplementary: Two angles sum to 180° (e.g., 130° + 50°).

Degrees, Minutes, and Seconds

• 1 degree = 60 minutes ('); 1 minute = 60 seconds (").
• To convert D° M' to decimal: degrees + (minutes/60).
• To convert decimal degrees to D° M': multiply the decimal by 60.

Radian Measure

• Radian is the angle measure where the arc length equals the radius.
• 1 full rotation = 2π radians = 360°.
• Radian formula: θ = s/r, where s = arc length, r = radius.

Converting Degrees and Radians

• Degrees to radians: multiply by (π/180).
• Radians to degrees: multiply by (180/π).

Coterminal Angles

• Coterminal angles differ by multiples of 360° or 2π radians.
• To find coterminal angles, add or subtract 360° (or 2π radians) as needed.

Arc Length and Applications

• Arc length formula: s = rθ (θ in radians).
• Convert degrees to radians before using the arc formula.
• To find arc length using latitude on Earth, subtract latitudes and convert to radians.

Linear and Angular Speed

• Angular speed (ω) = θ / t (radians per unit time).
• Linear speed (v) = s / t or v = rω.
• For wheels: ω in radians/minute, radius in feet gives linear speed in feet/minute.

Area of a Sector

• Area of sector: A = (1/2) r²θ (θ in radians).
• Convert the angle to radians before using the formula.

Key Terms & Definitions

• Initial Side — Starting position of an angle.
• Terminal Side — Position after rotation.
• Standard Position — Vertex at origin, initial side on positive x-axis.
• Right Angle — 90° angle.
• Straight Angle — 180° angle.
• Acute Angle — Less than 90°.
• Obtuse Angle — Between 90° and 180°.
• Complementary Angles — Angles summing to 90°.
• Supplementary Angles — Angles summing to 180°.
• Radian — Angle with arc length equal to radius.
• Coterminal Angles — Angles with same initial and terminal sides.
• Arc Length (s) — Distance along the circle, s = rθ.
• Angular Speed (ω) — Rate of angle change, ω = θ/t.
• Linear Speed (v) — Rate of arc length change, v = rω.

Action Items / Next Steps

• Practice converting between degrees, minutes, and decimal form.
• Practice converting between degrees and radians.
• Complete homework problems involving arc length, speed, and sector area.