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.