Coterminal Angles and Arc Length

This section covers the relationship between coterminal angles and arc length calculations on a circle.

Definition: Coterminal angles are angles that share the same terminal side and differ by 360° or 2π radians.

Example: Finding a coterminal angle for 960°: 960° - 360° = 600° - 360° = 240°

Highlight: Arc length (S) is calculated using the formula S = rθ, where r is the radius and θ is the angle in radians.

Example: For a circle with radius 15cm and angle 105°, the arc length is: S = 15 × $105π/180$ = 27.5cm

Linear and Angular Speed

This section explores the relationship between linear and angular speeds in circular motion.

Definition: Angular speed (ω) measures the rate of angular displacement, while linear speed $v$ measures the actual distance traveled per unit time.

Example: For a ceiling fan rotating at 90 rpm with a 2-foot blade: Angular speed = 180π rad/min Linear speed = 1131 ft/min

Highlight: The relationship between linear and angular speed is given by v = rω, where r is the radius.

Area of Sectors and Special Applications

This section details the calculation of sector areas and introduces advanced trigonometric concepts.

Definition: The area of a sector is calculated using the formula A = ½r²θ, where θ is in radians.

Example: For a sprinkler covering 150° with a 90-foot radius: A = ½(90)²$150π/180$ = 10,603 ft²

Highlight: Converting angles to radians is crucial for accurate sector area calculations.

DMS and Degree Decimal Conversions

This section explains the fundamental concepts of converting between DMS (Degrees, Minutes, Seconds) and decimal degree formats, as well as transformations between degrees and radians.

Definition: DMS (Degrees, Minutes, Seconds) is a format where 1 degree equals 60 minutes, and 1 minute equals 60 seconds.

Example: Converting 74°42'15" to decimal form: 74° + 42/60° + 15/3600° = 74.7042°

Highlight: When converting from decimal degrees to DMS, multiply the decimal portion by 60 repeatedly to obtain minutes and seconds.

Vocabulary: Radians are the standard unit for angle measurement in calculus, representing the ratio of arc length to radius.