Angles and circular motion are key concepts in math, linking geometry and trigonometry. They help us understand how objects move in circles and how to measure rotations, which is useful in many real-world applications.

We'll look at different ways to measure angles, convert between units, and calculate arc lengths. We'll also explore circular motion, including linear and angular velocity, and how these concepts relate to the unit circle and trigonometry.

Angles and Circular Motion

Angles in standard position

• Vertex of angle positioned at origin with initial side extending along positive x-axis
• Terminal side determined by angle's measure
  • Positive angles measured counterclockwise from initial side (90°, 180°)
  • Negative angles measured clockwise from initial side (-45°, -120°)
• Quadrantal angles have terminal side along coordinate axes
  • 0° or 0 radians: points along positive x-axis
  • 90° or $\frac{\pi}{2}$ radians: points along positive y-axis
  • 180° or $\pi$ radians: points along negative x-axis
  • 270° or $\frac{3\pi}{2}$ radians: points along negative y-axis

Degree and radian conversions

• Radians provide alternative unit for measuring angles
  • One radian equals angle subtended by arc length equal to radius of circle
• Converting degrees to radians: multiply angle in degrees by $\frac{\pi}{180}$
  • 60° = 60 × $\frac{\pi}{180}$ = $\frac{\pi}{3}$ radians
  • 135° = 135 × $\frac{\pi}{180}$ = $\frac{3\pi}{4}$ radians
• Converting radians to degrees: multiply angle in radians by $\frac{180}{\pi}$
  • $\frac{\pi}{6}$ radians = $\frac{\pi}{6}$ × $\frac{180}{\pi}$ = 30°
  • $\frac{5\pi}{4}$ radians = $\frac{5\pi}{4}$ × $\frac{180}{\pi}$ = 225°

Coterminal angles

• Angles sharing the same terminal side
• Finding coterminal angles in degrees: add or subtract multiples of 360° to given angle
  • 45° and 405° are coterminal (45° + 360° = 405°)
  • -30° and 330° are coterminal (-30° + 360° = 330°)
• Finding coterminal angles in radians: add or subtract multiples of 2$\pi$ to given angle
  • $\frac{\pi}{3}$ and $\frac{7\pi}{3}$ are coterminal ($\frac{\pi}{3}$ + 2$\pi$ = $\frac{7\pi}{3}$)
  • $-\frac{\pi}{4}$ and $\frac{7\pi}{4}$ are coterminal ($-\frac{\pi}{4}$ + 2$\pi$ = $\frac{7\pi}{4}$)

Arc length calculation

• Arc length proportional to measure of central angle
• Formula: $s = r\theta$
  • $s$: arc length
  • $r$: radius of circle
  • $\theta$: central angle in radians (convert from degrees if necessary)
• Circle with radius 8 units and central angle of 90°
  • Convert angle to radians: 90° × $\frac{\pi}{180}$ = $\frac{\pi}{2}$ radians
  • Calculate arc length: $s = 8 \times \frac{\pi}{2} = 4\pi \approx 12.57$ units
• Circle with radius 3 units and central angle of $\frac{2\pi}{3}$ radians
  • Calculate arc length: $s = 3 \times \frac{2\pi}{3} = 2\pi \approx 6.28$ units

Circular motion concepts

• Linear velocity ($v$): speed of object moving along circular path
  • Formula: $v = \frac{2\pi r}{T}$
    • $r$: radius of circle
    • $T$: period (time for one complete revolution)
• Angular velocity ($\omega$): rate of change of angular position
  • Formulas: $\omega = \frac{2\pi}{T}$ or $\omega = \frac{v}{r}$
  • Measured in radians per unit time
• Relationship between linear and angular velocity: $v = r\omega$
• Ferris wheel with radius 20 m and period of 30 seconds
  • Calculate linear velocity: $v = \frac{2\pi \times 20}{30} \approx 4.19$ m/s
  • Calculate angular velocity: $\omega = \frac{2\pi}{30} \approx 0.21$ rad/s
• Car wheel with radius 0.3 m and angular velocity of 10 rad/s
  • Calculate linear velocity: $v = 0.3 \times 10 = 3$ m/s

Trigonometry and the Unit Circle

• Trigonometry: branch of mathematics dealing with relationships between sides and angles of triangles
• Unit circle: circle with radius 1 centered at the origin, used to define trigonometric functions
• Rotation: angular movement around a fixed point, measured in degrees or radians
• Revolution: complete 360° rotation around a fixed point
• Circumference: distance around a circle, calculated using the formula C = 2πr, where r is the radius
• Pi (π): mathematical constant representing the ratio of a circle's circumference to its diameter, approximately 3.14159