Study Guide for Exam 4

This will be a closed-book exam. No calculators are permitted on this exam.

To do well on the exam, you should be able to do the following:

Section 5.1: The Unit Circle

  1. State the equation for the unit circle, and relate the equation to the distance formula we learned in Chapter 1.
  2. Define the terminal point \((x, y)\) for a given real number \(t\).
  3. State the terminal points for \(t = k \cdot \frac{\pi}{6}\), \(t = k \cdot \frac{\pi}{4}\), \(t = k \cdot \frac{\pi}{3}\), and \(t = k \cdot \frac{\pi}{2}\) where \(k\) is a given integer.
  4. Determine the reference number \(\bar{t}\) for a given real number \(t\).
  5. Use the procedure in the grey box on page 406 of Stewart, Redlin, and Watson to find the terminal point for any real number \(t\) using \(t\)’s reference number \(\bar{t}\).

Section 5.2: Trigonometric Functions of Real Numbers

  1. Relate the trigonometric functions sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)) to the terminal point associated with a real number \(t\).
  2. Evaluate the trigonmetric functions above at \(t = k \cdot \frac{\pi}{6}\), \(t = k \cdot \frac{\pi}{4}\), \(t = k \cdot \frac{\pi}{3}\), and \(t = k \cdot \frac{\pi}{2}\) where \(k\) is a given integer.
  3. Given a terminal point determined by a real number \(t\), determine the values of the trigonometric functions above evaluated at \(t\).
  4. State the signs of sine, cosine, and tangent in each of the 4 quadrants of the coordinate plane.
  5. Use the procedure in the grey box on page 412 of Stewart, Redlin, and Watson to evaluate the trigonometric functions at any real number.

Section 5.3: Trigonometric Graphs

  1. Relate the graphs of \(\sin t\) and \(\cos t\) to the terminal point \((x, y)\) a distance \(t\) from the point \((1, 0)\) along the unit circle.
  2. State what property a function must have to be periodic.
  3. State the period of \(\sin t\) and \(\cos t\).
  4. Sketch one period of the graphs of \(\sin t\) and \(\cos t\).
  5. Identify the amplitude and period of the functions \(s(t) = a \sin k(t - h)\) and \(c(t) = a \cos k(t - h)\).
  6. Explain the impacts of \(a\), \(k\), \(v\), and \(h\) on the transformations \(s(t) = v + a \sin k(t - h)\) and \(c(t) = v + a \cos k(t - h)\) of \(\sin t\) and \(\cos t\), and how they impact the graphs of the transformations.
  7. Identify the functions \(s(t) = v + a \sin k(t - h)\) and \(c(t) = v + a \cos k(t - h)\) from a graph of a single period.
  8. Identify the domain and range of the functions \(\sin t\), \(\cos t\), \(s(t) = v + a \sin k(t - h)\), and \(c(t) = v + a \cos k(t - h)\).

Section 5.6: Modeling Harmonic Motion

  1. State the equation for simple harmonic motion, and identify the amplitude, period, and frequency given the equation for simple harmonic motion.
  2. Relate the angular frequency \(k\) of a sine wave to its temporal frequency, which we will denote by either \(\nu\) (the lowercase Greek letter nu) or \(f\). Your textbook calls temporal frequency simply ‘frequency’.
  3. Identify the unit of frequency in the International System of Units.
  4. State the equation for simple harmonic motion.
  5. State the model for the displacement of a mass on a spring from its equilibrium position and identify the period, frequency, and amplitude of the mass’s motion.
  6. Identify a model for a pure tone (sound), and identify how its parameters are related to loudness and tone.
  7. Relate the frequency of a tone to the frequency of its octaves.