Chapter 4.2 Practice

Questions

  1. Identify the amplitude of the following functions.
    1. \(f(x)=\sin(x)\)
    2. \(f(x)=\cos(x)\)
    3. \(f(x)=-\sin(x)+4\)
    4. \(f(x)=5\sin(2x)\)
    5. \(f(x)=-2\cos(x+\pi)-1\)
  2. Determine the fundamental period of the following functions.
    1. \(f(x)=\sin(x)\)
    2. \(f(x)=\cos(x)\)
    3. \(f(x)=\tan(x)\)
    4. \(f(x)=2\sin(x+1)\)
    5. \(f(x)=\cos(\pi x)-1\)
    6. \(f(x)=2\tan(4x)\)
    7. \(f(x)=2\sin(2x-5)+1\)
    8. \(f(x)=3\cos(2\pi x)+1\)
    9. \(f(x)=\tan(\pi x+1)\)
  3. Take \(f\) to be a function with fundamental period equal to \(6\pi\) and take \(g\) to be given by \[g(x) = 4x - 5.\] Determine the fundamental period of \(f\circ g\).
  4. The sketch of a periodic function \(f\) is given below. Identify its fundamental period.

  1. Determine the phase shift of the following functions.
    1. \(f(x)=\sin(x)\)
    2. \(f(x)=\cos(x)\)
    3. \(f(x)=\sin(x-2)\)
    4. \(f(x)=\cos(2x+1)\)
  2. Graph the following functions. Identify the fundamental period, amplitude, domain and range.
    1. \(f(x)=3\sin\left(x-\frac{\pi}{2}\right)\)
    2. \(f(x)=4\cos(2\pi x-1)+2\)
  3. Whenever possible, evaluate the function \(\csc\), \(\sec\) and \(\cot\) at the angles measured in radians.
    1. \(0\)
    2. \(\frac{\pi}{6}\)
    3. \(\frac{\pi}{4}\)
    4. \(\frac{\pi}{3}\)
    5. \(\frac{\pi}{2}\)
    6. \(\frac{2\pi}{3}\)
    7. \(\frac{3\pi}{4}\)
    8. \(\frac{5\pi}{6}\)
    9. \(\pi\)
    10. \(\frac{7\pi}{6}\)
    11. \(\frac{5\pi}{4}\)
    12. \(\frac{4\pi}{3}\)
    13. \(\frac{3\pi}{2}\)
    14. \(\frac{5\pi}{3}\)
    15. \(\frac{7\pi}{4}\)
    16. \(\frac{11\pi}{6}\)
    17. \(2\pi\)
  4. Whenever possible, evaluate the function \(\csc\), \(\sec\) and \(\cot\) at the angles measured in degrees.
    1. \(0^{\circ}\)
    2. \(30^{\circ}\)
    3. \(45^{\circ}\)
    4. \(60^{\circ}\)
    5. \(90^{\circ}\)
    6. \(120^{\circ}\)
    7. \(135^{\circ}\)
    8. \(150^{\circ}\)
    9. \(180^{\circ}\)
    10. \(210^{\circ}\)
    11. \(225^{\circ}\)
    12. \(240^{\circ}\)
    13. \(270^{\circ}\)
    14. \(300^{\circ}\)
    15. \(315^{\circ}\)
    16. \(330^{\circ}\)
    17. \(360^{\circ}\)
  5. Evaluate the function \(\arcsin\) and \(\arccos\) at the given values.
    1. \(-1\)
    2. \(-\frac{\sqrt{3}}{2}\)
    3. \(-\frac{\sqrt{2}}{2}\)
    4. \(-\frac{1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)
  6. Evaluate the function \(\arctan\) at the given values.
    1. \(-\sqrt{3}\)
    2. \(-1\)
    3. \(-\frac{1}{\sqrt{3}}\)
    4. \(0\)
    5. \(\frac{1}{\sqrt{3}}\)
    6. \(1\)
    7. \(\sqrt{3}\)
  7. Calculate the following.
    1. \(\arcsin\left(\sin\left(-\frac{\pi}{3}\right)\right)\)
    2. \(\arctan\left(\tan\left(\frac{\pi}{5}\right)\right)\)
    3. \(\arccos\left(\cos\left(\frac{\pi}{15}\right)\right)\)
    4. \(\sin\left(\arcsin\left(\frac{1}{2}\right)\right)\)
    5. \(\cos\left(\arccos\left(-\frac{1}{3}\right)\right)\)
    6. \(\tan\left(\arctan\left(\sqrt{3}\right)\right)\)
    7. \(\arccos\left(\cos\left(\frac{7\pi}{5}\right)\right)\)
    8. \(\arcsin\left(\sin\left(\frac{5\pi}{7}\right)\right)\)
    9. \(\arctan\left(\tan\left(\frac{9\pi}{5}\right)\right)\)
  8. Calculate the following.
    1. \(\cos\left(\arctan\left(\frac{2}{7}\right)\right)\)
    2. \(\sin\left(\arccos\left(-\frac{5}{9}\right)\right)\)
    3. \(\tan\left(\arcsin\left(\frac{1}{11}\right)\right)\)
  9. Solve the following equations.
    1. \(\cos(x)=\frac{1}{2}\) on \([0,2\pi)\)
    2. \(\cos^2(x)-\frac{1}{4}=0\) on \(\mathbb{R}\)
    3. \(\sin(x)-1=0\) on \([0,2\pi)\)
    4. \(3\cos(x)-3=0\) on \(\mathbb{R}\)
    5. \(\sqrt{3}\tan(x)=0\) on \([0,2\pi)\)
    6. \(\tan(x)+\sqrt{3}=0\) on \(\mathbb{R}\)
    7. \(\cos^2(\theta)+\cos(\theta)-\frac{3}{4}=0\) on \(\mathbb{R}\)
    8. \(\cos(x)-\frac{1}{3}=0\) on \(\mathbb{R}\)
    9. \(\sin(x)=0.45\) on \(\mathbb{R}\)
    10. \(\sin(x)=5\)
    11. \(\cos(5x)=\frac{1}{2}\) on \(\mathbb{R}\)
  10. A point \(\left(x,-\frac{6}{11}\right)\) on the unit circle corresponds to an angle \(\theta\) in quadrant III. Calculate \[\sin(\theta),\quad \cos(\theta),\quad \tan(\theta),\quad \csc(\theta),\quad \cot(\theta)\quad \text{and} \quad\sec(\theta).\]

Answers

    1. \(1\)
    2. \(1\)
    3. \(1\)
    4. \(5\)
    5. \(-2\)
    1. \(2\pi\)
    2. \(2\pi\)
    3. \(\pi\)
    4. \(\pi\)
    5. \(2\)
    6. \(\frac{\pi}{4}\)
    7. \(\pi\)
    8. \(1\)
    9. \(1\)
  3. \(\frac{3\pi}{2}\)
  4. \(4\)
    1. \(0\)
    2. \(0\)
    3. \(\frac{1}{\pi}\)
    4. \(-\frac{1}{2\pi}\)
    1. The fundamental period is \(2\pi\), amplitude is \(3\), domain is \(\mathbb{R}\) and the range is \([-3,3]\). The graph of \(f\) is given below.
    2. The fundamental period is \(1\), amplitude is \(4\), domain is \(\mathbb{R}\) and the range is \([-2,6]\). The graph of \(f\) is given below.
    1. \(\sec(0)=1\)
    2. \(\csc\left(\frac{\pi}{6}\right)=2\), \(\sec\left(\frac{\pi}{6}\right)=\frac{2}{\sqrt{3}}\), \(\cot\left(\frac{\pi}{6}\right)=\sqrt{3}\)
    3. \(\csc\left(\frac{\pi}{4}\right)=\sqrt{2}\), \(\sec\left(\frac{\pi}{4}\right)=\sqrt{2}\), \(\cot\left(\frac{\pi}{4}\right)=1\)
    4. \(\csc\left(\frac{\pi}{3}\right)=\frac{2}{\sqrt{3}}\), \(\sec\left(\frac{\pi}{3}\right)=2\), \(\cot\left(\frac{\pi}{3}\right)=\frac{1}{\sqrt{3}}\)
    5. \(\csc\left(\frac{\pi}{2}\right)=1\), \(\cot\left(\frac{\pi}{2}\right)=0\)
    6. \(\csc\left(\frac{2\pi}{3}\right)=\frac{2}{\sqrt{3}}\), \(\sec\left(\frac{2\pi}{3}\right)=2\), \(\cot\left(\frac{2\pi}{3}\right)=-\frac{1}{\sqrt{3}}\)
    7. \(\csc\left(\frac{3\pi}{4}\right)=\sqrt{2}\), \(\sec\left(\frac{3\pi}{4}\right)=-\sqrt{2}\), \(\cot\left(\frac{3\pi}{4}\right)=-1\)
    8. \(\csc\left(\frac{5\pi}{6}\right)=2\), \(\sec\left(\frac{5\pi}{6}\right)=-\frac{2}{\sqrt{3}}\), \(\cot\left(\frac{5\pi}{6}\right)=-\sqrt{3}\)
    9. \(\sec(\pi)=-1\)
    10. \(\csc\left(\frac{7\pi}{6}\right)=-2\), \(\sec\left(\frac{7\pi}{6}\right)=-\frac{2}{\sqrt{3}}\), \(\cot\left(\frac{7\pi}{6}\right)=\sqrt{3}\)
    11. \(\csc\left(\frac{5\pi}{4}\right)=-\sqrt{2}\), \(\sec\left(\frac{5\pi}{4}\right)=-\sqrt{2}\), \(\cot\left(\frac{5\pi}{4}\right)=1\)
    12. \(\csc\left(\frac{4\pi}{3}\right)=-\frac{2}{\sqrt{3}}\), \(\sec\left(\frac{4\pi}{3}\right)=-2\), \(\cot\left(\frac{4\pi}{3}\right)=\frac{1}{\sqrt{3}}\)
    13. \(\csc\left(\frac{3\pi}{2}\right)=-1\), \(\cot\left(\frac{3\pi}{2}\right)=0\)
    14. \(\csc\left(\frac{5\pi}{3}\right)=-\frac{2}{\sqrt{3}}\), \(\sec\left(\frac{5\pi}{3}\right)=2\), \(\cot\left(\frac{5\pi}{3}\right)=-\frac{1}{\sqrt{3}}\)
    15. \(\csc\left(\frac{7\pi}{4}\right)=-\sqrt{2}\), \(\sec\left(\frac{7\pi}{4}\right)=\sqrt{2}\), \(\cot\left(\frac{7\pi}{4}\right)=-1\)
    16. \(\csc\left(\frac{11\pi}{6}\right)=-2\), \(\sec\left(\frac{11\pi}{6}\right)=\frac{2}{\sqrt{3}}\), \(\cot\left(\frac{11\pi}{6}\right)=-\sqrt{3}\)
    17. \(\sec(2\pi)=1\)
    1. \(\sec(0^{\circ})=1\)
    2. \(\csc\left(30^{\circ}\right)=2\), \(\sec\left(30^{\circ}\right)=\frac{2}{\sqrt{3}}\), \(\cot\left(30^{\circ}\right)=\sqrt{3}\)
    3. \(\csc\left(45^{\circ}\right)=\sqrt{2}\), \(\sec\left(45^{\circ}\right)=\sqrt{2}\), \(\cot\left(45^{\circ}\right)=1\)
    4. \(\csc\left(60^{\circ}\right)=\frac{2}{\sqrt{3}}\), \(\sec\left(60^{\circ}\right)=2\), \(\cot\left(60^{\circ}\right)=\frac{1}{\sqrt{3}}\)
    5. \(\csc\left(90^{\circ}\right)=1\), \(\cot\left(90^{\circ}\right)=0\)
    6. \(\csc\left(120^{\circ}\right)=\frac{2}{\sqrt{3}}\), \(\sec\left(120^{\circ}\right)=2\), \(\cot\left(120^{\circ}\right)=-\frac{1}{\sqrt{3}}\)
    7. \(\csc\left(135^{\circ}\right)=\sqrt{2}\), \(\sec\left(135^{\circ}\right)=-\sqrt{2}\), \(\cot\left(135^{\circ}\right)=-1\)
    8. \(\csc\left(150^{\circ}\right)=2\), \(\sec\left(150^{\circ}\right)=-\frac{2}{\sqrt{3}}\), \(\cot\left(150^{\circ}\right)=-\sqrt{3}\)
    9. \(\sec(180^{\circ})=-1\)
    10. \(\csc\left(210^{\circ}\right)=-2\), \(\sec\left(210^{\circ}\right)=-\frac{2}{\sqrt{3}}\), \(\cot\left(210^{\circ}\right)=\sqrt{3}\)
    11. \(\csc\left(225^{\circ}\right)=-\sqrt{2}\), \(\sec\left(225^{\circ}\right)=-\sqrt{2}\), \(\cot\left(225^{\circ}\right)=1\)
    12. \(\csc\left(240^{\circ}\right)=-\frac{2}{\sqrt{3}}\), \(\sec\left(240^{\circ}\right)=-2\), \(\cot\left(240^{\circ}\right)=\frac{1}{\sqrt{3}}\)
    13. \(\csc\left(270^{\circ}\right)=-1\), \(\cot\left(270^{\circ}\right)=0\)
    14. \(\csc\left(300^{\circ}\right)=-\frac{2}{\sqrt{3}}\), \(\sec\left(300^{\circ}\right)=2\), \(\cot\left(300^{\circ}\right)=-\frac{1}{\sqrt{3}}\)
    15. \(\csc\left(315^{\circ}\right)=-\sqrt{2}\), \(\sec\left(315^{\circ}\right)=\sqrt{2}\), \(\cot\left(315^{\circ}\right)=-1\)
    16. \(\csc\left(330^{\circ}\right)=-2\), \(\sec\left(330^{\circ}\right)=\frac{2}{\sqrt{3}}\), \(\cot\left(330^{\circ}\right)=-\sqrt{3}\)
    17. \(\sec(360^{\circ})=1\)
    1. \(\arcsin(-1)=-\frac{\pi}{2}\), \(\arccos(-1)=\pi\)
    2. \(\arcsin\left(-\frac{\sqrt{3}}{2}\right)=-\frac{\pi}{3}\), \(\arccos\left(-\frac{\sqrt{3}}{2}\right)=\frac{5\pi}{6}\)
    3. \(\arcsin\left(-\frac{\sqrt{2}}{2}\right)=-\frac{\pi}{4}\), \(\arccos\left(-\frac{\sqrt{2}}{2}\right)=\frac{3\pi}{4}\)
    4. \(\arcsin\left(-\frac{1}{2}\right)=-\frac{\pi}{6}\), \(\arccos\left(-\frac{1}{2}\right)=\frac{2\pi}{3}\)
    5. \(\arcsin(0)=0\), \(\arccos(0)=\frac{\pi}{2}\)
    6. \(\arcsin\left(\frac{1}{2}\right)=\frac{\pi}{6}\), \(\arccos\left(\frac{1}{2}\right)=\frac{\pi}{3}\)
    7. \(\arcsin\left(\frac{\sqrt{2}}{2}\right)=\frac{\pi}{4}\), \(\arccos\left(\frac{\sqrt{2}}{2}\right)=\frac{\pi}{4}\)
    8. \(\arcsin\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}\), \(\arccos\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{6}\)
    9. \(\arcsin(1)=\frac{\pi}{2}\), \(\arccos(1)=0\)
    1. \(-\frac{\pi}{3}\)
    2. \(-\frac{\pi}{4}\)
    3. \(-\frac{\pi}{6}\)
    4. \(0\)
    5. \(\frac{\pi}{6}\)
    6. \(\frac{\pi}{4}\)
    7. \(\frac{\pi}{3}\)
  11. Calculate the following.
    1. \(-\frac{\pi}{3}\)
    2. \(\frac{\pi}{5}\)
    3. \(\frac{\pi}{15}\)
    4. \(\frac{1}{2}\)
    5. \(-\frac{1}{3}\)
    6. \(\sqrt{3}\)
    7. \(\frac{3\pi}{5}\)
    8. \(\frac{2\pi}{7}\)
    9. \(-\frac{\pi}{5}\)
    1. \(\frac{7}{\sqrt{53}}\)
    2. \(\frac{\sqrt{56}}{9}\)
    3. \(\frac{\sqrt{30}}{60}\)
    1. \(\frac{\pi}{3},\frac{5\pi}{3}\)
    2. \(\frac{\pi}{3}+2\pi k,\frac{2\pi}{3}+2\pi k,\frac{4\pi}{3}+2\pi k, \frac{5\pi}{3}+2\pi k\) where \(k\) is an integer
    3. \(\frac{\pi}{2}\)
    4. \(2\pi k\) where \(k\) is an integer
    5. \(\frac{\pi}{6},\frac{11\pi}{6}\)
    6. \(\frac{2\pi}{3}+\pi k\), \(\frac{5\pi}{3}+\pi k\) or just \(\frac{2\pi}{3}+\pi k\) where \(k\) is an integer
    7. \(\frac{\pi}{3}+2\pi k, \frac{5\pi}{3}+2\pi k\) where \(k\) is an integer
    8. \(\arccos\left(\frac{1}{3}\right)+2\pi k\), \(2\pi-\arccos\left(\frac{1}{3}\right)+2\pi k\) where \(k\) is an integer
    9. \(\arcsin(0.45)+2\pi k\), \(\pi-\arcsin(0.45)+2\pi k\) where \(k\) is an integer
    10. no solution
    11. \(\frac{\pi}{15}+\frac{2\pi}{5}k\), \(\frac{1\pi}{3}+\frac{2\pi}{5}k\) where \(k\) is an integer
  14. \(\sin(\theta)=-\frac{6}{11}\), \(\cos(\theta)=-\frac{\sqrt{85}}{11}\), \(\tan(\theta)=\frac{6}{\sqrt{85}}\), \(\csc(\theta)=-\frac{11}{6}\), \(\cot(\theta)=\frac{\sqrt{85}}{6}\), \(\sec(\theta)=-\frac{11}{\sqrt{85}}.\)

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