MAT-170 Precalculus Exam 3 Review

Circular Motion and Trigonometry

• Angle Sweeping and Distance Travelled
  • Circular Track Problem
  • Radius of circular track = 1.3 miles
  • Distance travelled when angle is 3.4 radians:
    • Formula: Distance = Radius * Angle (in radians)
    • Calculation: ext{Distance} = 1.3 imes 3.4
    • Result: 4.42 miles
  • Angle Swept when distance is 6.3 miles:
    • Formula: Angle = Distance / Radius
    • Calculation: ext{Angle} = rac{6.3}{1.3}
    • Result: 4.85 radians
  • Vertical Distance Above Center:
    • Function: f(s) = R imes ext{sin}igg(rac{s}{R}igg)
    • Where:
    • R = 1.3 (radius)
    • s = Distance travelled
  • Vertical Distance for angle 2.1 radians:
    • Using function: f(2.1) = 1.3 imes ext{sin}(2.1)
    • Result: approx. 1.03 miles
  • Horizontal Distance for 2 miles travelled:
    • Calculate angle first heta = rac{2}{1.3}
    • Use cos: d_{x} = R imes ext{cos} heta

Ferris Wheel Dynamics

• Michael's Ferris Wheel Problem
  • Radius = 35 feet, starting position = 3 o'clock, bottom = 5 feet above ground
  • Arc Length for 150 degrees:
  • Convert degrees to radians: ext{radians} = rac{150 imes ext{π}}{180} = rac{5 ext{π}}{6}
  • Distance travelled = Radius * Angle in radians: Distance = 35 imes rac{5 ext{π}}{6}
  • Result: Approx. 29.54 feet
  • Arc Length for 22 feet:
  • Find angle in radians: ext{angle in radians} = rac{22}{35}
  • Convert to degrees: ext{degrees} = rac{22}{35} imes rac{180}{ ext{π}}
  • Vertical Height After 22 feet:
  • Use h = 5 + 35 imes ext{sin}igg(rac{22}{35}igg)

Definition of Functions for Vertical Distances

• Function Definitions
  • Michael's vertical distance above ground:
  • f(θ) = 5 + R imes ext{sin}(θ)
  • Where θ in radians
  • Ferris wheel completes 3 revolutions in 55 mins:
  • Radians per minute: ext{radians/min} = rac{6 ext{π}}{55}
  • Angle function in terms of time f(t) = rac{6 ext{π}}{55}t
  • Vertical distance function:
  • g(t) = 5 + 35 imes ext{sin}(f(t))

Coordinates and Slope of Angles

• Coordinates Calculation
  • Determine position of terminal ray and slope:
  • Point Coordinates:
    • x = R imes ext{cos}(θ), y = R imes ext{sin}(θ)
  • Slope: rac{y}{x}

Trigonometric Functions Overview

• Functions and Variability
  • Behavior of sin, cos, tan as angles change:
  • From 0 to rac{ ext{π}}{2}, ext{sin}(θ) increases from 0 to 1.
  • Angle Conversion:
    • 140 degrees to radians: rac{140 ext{π}}{180}
    • 13π/10 to degrees: rac{13 ext{π}}{10} imes rac{180}{ ext{π}}
  • Hokies Conversion:
  • Questions on fractional representation and conversions involving Hokies.

Solving Trigonometric Equations

• Various Problems
  • Compute height of triangles using sin, cos measures.
  • Evaluate identities and apply in real-world contexts, such as angle of elevation and distances using trigonometric ratios.