Chapter 4 Section 2

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Identify the amplitude, fundamental period and phase shift of the following functions: \[f(x)=-\sin(\pi x+3)-1.\]
  2. Take \(f\) to be a function with fundamental period equal to \(\frac{1}{2}\) and take \(g\) to be given by \[g(x) = 7x + \pi.\] Determine the fundamental period of \(f\circ g\).
  3. Graph the following function and identify the fundamental period, amplitude, domain and range. \[f(x)=-\sin(\pi x+3)-1.\]
  4. Calculate the following.
    1. \(\cot\left(\frac{7\pi}{6}\right)\)
    2. \(\csc(\theta)\) given that \(\sin(\theta)=-\frac{1}{9}\)
    3. \(\arccos\left(-\frac{\sqrt{3}}{2}\right)\)
    4. \(\arcsin\left(-\frac{\sqrt{3}}{2}\right)\)
    5. \(\arctan\left(-\sqrt{3}\right)\)
  5. Calculate the following.
    1. \(\arccos\left(\cos\left(\frac{5\pi}{7}\right)\right)\)
    2. \(\arcsin\left(\sin\left(\frac{3\pi}{4}\right)\right)\)
    3. \(\arctan\left(\tan\left(\frac{5\pi}{3}\right)\right)\)
    4. \(\sin\left(\arctan\left(\frac{3}{5}\right)\right)\)
  6. Solve the following equations.
    1. \(\cos(x)=-\frac{\sqrt{2}}{2}\) on \([0,2\pi)\)
    2. \(5 \sin(x)-\frac{5}{2}=0\) on \(\mathbb{R}\)
    3. \(\cos^2(\theta) + \frac{3}{2}\cos(\theta) - 1=0\) on \(\mathbb{R}\)
    4. \(\tan(x)-\frac{2}{3}=0\) on \(\mathbb{R}\)
    5. \(\cos(x)=-1.1\)
  7. A point \(\left(x,-\frac{3}{5}\right)\) on the unit circle corresponds to an angle \(\theta\) in quadrant III. Calculate \[\sin(\theta),\quad \cos(\theta),\quad \tan(\theta),\quad \csc(\theta),\quad \cot(\theta)\quad \text{and} \quad\sec(\theta).\]

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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