Angles are the building blocks of trigonometry, and measuring them is key. We'll look at degrees and radians, two ways to describe angles, and how to switch between them. We'll also see how angles relate to circular motion and arc length.

Understanding angles helps us tackle real-world problems. We'll explore how to find related angles, calculate speeds in circular motion, and apply these concepts to things like rotating machinery and planetary orbits. It's all about connecting math to the world around us.

Angle Measurement and Circular Motion

Degree and radian conversion

• Radian measure defined as angle subtended by arc length equal to radius
• Relationship between degrees and radians $360° = 2π$ radians, $180° = π$ radians
• Conversion formulas: $θ_{rad} = θ_{deg} × \frac{π}{180°}$, $θ_{deg} = θ_{rad} × \frac{180°}{π}$
• Common angle measures: $90° = \frac{π}{2}$ rad, $45° = \frac{π}{4}$ rad, $30° = \frac{π}{6}$ rad

Arc length and central angle relationship

• Arc length formula: $s = rθ$ (s: arc length, r: radius, θ: central angle in radians)
• Formula components interpretation reveals proportional relationship
• Applications: find missing values (arc length, central angle, radius)
• Ratio of arc length to radius equals radian measure of central angle

Coterminal and reference angles

• Coterminal angles share terminal side, differ by multiples of 360° or 2π rad
• Find coterminal angles by adding/subtracting 360° or 2π rad
• Reference angles: acute angle between terminal side and x-axis
• Calculate reference angles for standard position (0° to 360°) and beyond
• Reference angles relate to trigonometric function values

Angular speed in circular motion

• Angular speed: rate of change of central angle over time, $ω = \frac{θ}{t}$
• Convert between angular and linear speed: $v = rω$ (r: radius of circular path)
• Solve rotation problems involving periods (time for one revolution) and frequencies (revolutions per unit time)
• Real-world applications (rotating machinery, planetary motion, clock hands)