Review Exercises - Trigonometry

  1. Find the values of the following:

    1. $\displaystyle{\tan\,(-\frac{7\pi}{6})}$   $\ans{\displaystyle{-\frac{\sqrt{3}}{3}}}$

    2. $\displaystyle{\cot\, \theta, \textrm{ if } \sin\, \theta = \frac{2}{3} \textrm{ and } \frac{\pi}{2} \lt \theta \lt \pi}$   $\ans{\displaystyle{-\frac{\sqrt{5}}{2}}}$

    3. $\displaystyle{\cos\, (-\frac{11\pi}{6})}$   $\ans{\displaystyle{\frac{\sqrt{3}}{2}}}$

    4. $\displaystyle{\csc t, \textrm{ if } \cos t = -\frac{15}{17} \textrm{ and } \pi \lt t \lt \frac{3\pi}{2}}$   $\ans{\displaystyle{-\frac{17}{8}}}$

    5. $\displaystyle{\tan\, (-\frac{\pi}{6})}$   $\ans{\displaystyle{-\frac{\sqrt{3}}{3}}}$

    6. $\displaystyle{\arccos\, (\sin \frac{11\pi}{4})}$   $\ans{\displaystyle{\frac{\pi}{4}}}$

    7. $\displaystyle{\cos\, (\arctan\, (-1))}$   $\ans{\displaystyle{\frac{\sqrt{2}}{2}}}$

    8. $\displaystyle{\sec\, (\textrm{arccot}\, (-\frac{5}{12}))}$   $\ans{\displaystyle{-\frac{13}{5}}}$

    9. $\displaystyle{\textrm{arcsec}\, (-\sqrt{2})}$   $\ans{\displaystyle{\frac{3\pi}{4}}}$

    10. $\displaystyle{\sec\, (\textrm{arccot}\, \frac{x}{3})}$   $\ans{\displaystyle{\frac{\sqrt{x^2+9}}{x}}}$

    11. $\displaystyle{\arctan\,(\sin\, (-\frac{5\pi}{2}))}$   $\ans{\displaystyle{-\frac{\pi}{4}}}$

    1. $\displaystyle{\sec\, (-\frac{5\pi}{6})}$   $\ans{\displaystyle{-\frac{2}{\sqrt{3}}}}$

    2. $\displaystyle{9\,\textrm{arccot}^2\, \frac{\sqrt{3}}{3}}$   $\ans{\displaystyle{\pi^2}}$

    3. $\displaystyle{\textrm{arccsc}\, (-2)}$   $\ans{\displaystyle{-\frac{\pi}{6}}}$

    4. $\displaystyle{\tan\,(\arcsin\,(-\frac{3}{5}))}$   $\ans{\displaystyle{-\frac{3}{4}}}$

    5. $\displaystyle{\arccos\, (\cot \frac{11\pi}{4})}$   $\ans{\displaystyle{\pi}}$

    6. $\displaystyle{\csc t, \textrm{ if } \tan t = 3 \textrm{ and } \pi \lt t \lt \frac{3\pi}{2}}$   $\ans{\displaystyle{-\frac{\sqrt{10}}{3}}}$

    7. $\displaystyle{\textrm{arccsc}\,(\cos\,(-\frac{\pi}{3}))}$   $\ans{\displaystyle{\textrm{no value}}}$

    8. $\displaystyle{\tan \theta, \textrm{ if } \sin \theta = \frac{5}{13} \textrm{ and } \frac{\pi}{2} \lt \theta \lt \pi}$   $\ans{\displaystyle{-\frac{5}{12}}}$

    9. $\displaystyle{\tan \frac{9\pi}{2}}$   $\ans{\displaystyle{\textrm{no value}}}$


  2. Find the values of the following:

    1. $\displaystyle{\cos\, (\arctan \frac{x}{3})}$   $\ans{\displaystyle{\frac{3}{\sqrt{x^2+9}}}}$

    2. $\displaystyle{\arcsin\, (\cos \frac{\pi}{2})}$   $\ans{\displaystyle{0}}$

    3. $\displaystyle{\textrm{arcsec}\, 1}$   $\ans{\displaystyle{0}}$

    4. $\displaystyle{\csc\, (-\frac{9\pi}{4})}$   $\ans{\displaystyle{-\sqrt{2}}}$

    5. $\displaystyle{\cos\,(\arcsin\, x)}$   $\ans{\displaystyle{\sqrt{1-x^2}}}$

    6. $\displaystyle{\tan\, \frac{54\pi}{6}}$   $\ans{\displaystyle{0}}$

    7. $\displaystyle{\textrm{arcsec}^2\, (\csc \frac{2\pi}{3})}$   $\ans{\displaystyle{\frac{\pi^2}{36}}}$

    8. $\displaystyle{\cos^2\, \frac{7\pi}{6}}$   $\ans{\displaystyle{\frac{3}{4}}}$

    9. $\displaystyle{\textrm{arccsc}\,(-\sqrt{2})}$   $\ans{\displaystyle{-\frac{\pi}{4}}}$

    10. $\displaystyle{5\, \textrm{arccot}\,(-\frac{\sqrt{3}}{3})}$   $\ans{\displaystyle{\frac{10\pi}{3}}}$

    11. $\displaystyle{\cot\, (\arcsin\, (-\frac{8}{17}))}$   $\ans{\displaystyle{-\frac{15}{8}}}$

    1. $\displaystyle{\arcsin\, (\sin\, \frac{5\pi}{3})}$   $\ans{\displaystyle{-\frac{\pi}{3}}}$

    2. $\displaystyle{\cos\, (\arctan\, 5)}$   $\ans{\displaystyle{\frac{\sqrt{26}}{26}}}$

    3. $\displaystyle{\arccos\, (\sin\, (-\frac{\pi}{6}))}$   $\ans{\displaystyle{\frac{2\pi}{3}}}$

    4. $\displaystyle{\sin t, \textrm{ if } \cot t = -\frac{12}{5} \textrm{ and } \frac{3\pi}{2} \lt t \lt 2\pi}$   $\ans{\displaystyle{-\frac{5}{13}}}$

    5. $\displaystyle{\cos\, (\arctan (-\frac{3}{4}))}$   $\ans{\displaystyle{\frac{4}{5}}}$

    6. $\displaystyle{\textrm{arcsec}\, (-\frac{2\sqrt{3}}{3})}$   $\ans{\displaystyle{}}$

    7. $\displaystyle{\arccos\, (\sin\, \frac{23\pi}{4})}$   $\ans{\displaystyle{\frac{5\pi}{6}}}$

    8. $\displaystyle{\arccos\, (\sin\, \frac{23\pi}{4})}$   $\ans{\displaystyle{\frac{\pi}{3}}}$

    9. $\displaystyle{\cot t, \textrm{ if } \sec t = -\frac{8}{5} \textrm{ and } \pi \lt t \lt \frac{3\pi}{2}}$   $\ans{\displaystyle{\frac{5\sqrt{39}}{39}}}$


  3. Sketch graphs of the following. Label intercepts, asymptotes, and endpoints. Also, give the amplitude, period, and phase shift.

    1. $\displaystyle{y=-4\, \cos\, (\frac{x}{2} - \frac{\pi}{4}) \quad \textrm{on} \quad [-\pi,3\pi]}$   $\ians{}$

    2. $\displaystyle{4\, \sin\, (\frac{2}{3} x + \frac{\pi}{6}) \quad \textrm{from} \quad -2\pi \, \textrm{ to } \, 2\pi}$   $\ians{}$

    3. $\displaystyle{y = 5\, \cos\, (3x + \frac{\pi}{2}) \quad \textrm{on} \quad [0,\pi]}$   $\ians{}$

    4. $\displaystyle{y = 4 - 2 \sin\, (\frac{x}{3} - \frac{\pi}{3}) \quad \textrm{from} \quad -2\pi \, \textrm{ to } \, \frac{5\pi}{2}}$   $\ians{}$

    5. $\displaystyle{y = -2\, \sin\, (x - \frac{3\pi}{2}) \quad \textrm{on} \quad [-\frac{\pi}{2}, 2\pi]}$   $\ians{}$

    6. $\displaystyle{y = -\frac{1}{3} \cos\, (\frac{x}{2} + \frac{\pi}{4}) \quad \textrm{on} \quad [-2\pi,4\pi]}$   $\ians{}$

  4. Sketch graphs of the following. Label intercepts, asymptotes, and endpoints.

    1. $\displaystyle{y = -2\sec\, (3x+\pi) \quad \textrm{on} \quad [-\frac{\pi}{3}, \frac{2\pi}{3}]}$   $\ians{}$

    2. $\displaystyle{y = 2\csc\, (2x + \frac{\pi}{2}) \quad \textrm{for} \quad -\frac{\pi}{2} \le x \le \pi }$   $\ians{}$

    3. $\displaystyle{y = \tan\, (x + \frac{\pi}{4}) \quad \textrm{on} \quad [-\pi, 2\pi]}$   $\ians{}$

    4. $\displaystyle{y = \cot\, (x - \frac{\pi}{6}) \quad \textrm{from} \quad -\pi \,\textrm{ to }\, 2\pi}$   $\ians{}$

    5. $\displaystyle{y = -\sec \frac{1}{2} x \quad \textrm{for} \quad -3\pi \lt x \lt 3\pi}$   $\ians{}$

  5. Show whether each of the following is or is not an identity:

    1. $\displaystyle{\frac{1}{\csc \theta + \cot \theta} + \frac{\sec \theta + 1}{\tan \theta} = 2 \csc \theta}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    2. $\displaystyle{\frac{1+\tan \theta}{1-\tan \theta} + \frac{1+\cot \theta}{1-\cot \theta} = 0}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    3. $\displaystyle{\frac{1}{1+\cos \theta} - \frac{1}{1 - \cos \theta} = \frac{2}{\sec \theta - \cos \theta}}$   $\ans{\displaystyle{\textrm{this is not an identity}}}$

    4. $\displaystyle{(1+\sec x)(1-\cos x) = \tan x \sin x}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    5. $\displaystyle{\sec 2\theta = \frac{\sec^2 \theta}{2 - \sec^2 \theta}}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    6. $\displaystyle{\frac{1+\sin \theta}{\cos \theta} + \frac{\cos \theta}{1+\sin \theta} = 2\sec \theta}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    7. $\displaystyle{\frac{\tan^2 x \csc^2 x - 1}{\csc x \, \tan^2 x \, \sin x} = 1}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    8. $\displaystyle{\frac{1}{\sec \theta - \tan \theta} = \sec \theta + \tan \theta}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    9. $\displaystyle{\frac{\sin \theta + \tan \theta}{1 + \cos \theta} = \cot \theta}$   $\ans{\displaystyle{\textrm{this is not an identity}}}$

    10. $\displaystyle{\cos^4 x - \sin^4 x = \cos 2x}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    11. $\displaystyle{\frac{\tan t}{\tan^2 t - 1} = \frac{1}{\tan t - \cot t}}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    12. $\displaystyle{\sec x - \sin x \tan x = \cos x}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    13. $\displaystyle{\frac{\sin \theta + \cos \theta}{\sec \theta + \csc \theta} = \frac{\sin \theta}{\sec \theta}}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    14. $\displaystyle{\frac{\cos^4 x - \sin^4 x}{\sin x + \cos x} = \frac{1-\tan x}{1+ \tan x}}$   $\ans{\displaystyle{\textrm{this is not an identity}}}$

  6. Find all solutions of the following equations:

    1. $\displaystyle{\tan^2 x + \sec^2 x + 3\sec x = 1}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{2\pi}{3} \pm 2\pi\,n\\\frac{4\pi}{3} \pm 2\pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    2. $\displaystyle{\cos 2x = 1+\sin x}$   $\ans{\displaystyle{\left. \begin{array}{ll}0 \pm \pi\,n\\\frac{7\pi}{6} \pm \pi\,n\\\frac{11\pi}{6} \pm \pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    3. $\displaystyle{2\sin x \, \tan x + \tan x - 2\sin x -1 = 0}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{7\pi}{6} \pm 2\pi\,n\\\frac{11\pi}{6} \pm 2\pi\,n\\\\\frac{\pi}{4} \pm 2\pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    4. $\displaystyle{2\cos^2 x - 3\cos x + 1 = 0}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{\pi}{3} \pm 2\pi\,n\\\frac{5\pi}{3} \pm 2\pi\,n\\\\0 \pm \pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    5. $\displaystyle{3\tan x + \frac{1}{\tan x} = 2\sqrt{3}}$   $\ans{\displaystyle{\frac{\pi}{6} \pm \pi\,n \textrm{ where } n=0,1,2,\ldots}}$

    6. $\displaystyle{2\sin x \, \tan x = 3}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{\pi}{3} \pm 2\pi\,n\\\frac{5\pi}{3} \pm 2\pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    7. $\displaystyle{\sec^2 x - \tan x = 1}$   $\ans{\displaystyle{\left. \begin{array}{ll}0 \pm \pi\,n\\\frac{\pi}{4} \pm \pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    8. $\displaystyle{4\sin^2 x - 1 = 0}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{\pi}{6} \pm \pi\,n\\\frac{5\pi}{6} \pm \pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    9. $\displaystyle{\cos 2x = \cos x}$   $\ans{\displaystyle{\begin{array}{l} \left. \begin{array}{ll}\frac{2\pi}{3} \pm 2\pi\,n\\\frac{4\pi}{3} \pm 2\pi\,n \\ 0 \pm 2\pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots\\\\
      \hline\\\\
      \textrm{more compactly: } \frac{2\pi\,n}{3} \textrm{ where } n=0,1,2,\ldots\end{array}}}$

    10. $\displaystyle{2\cos^3 x + \sin^2 x = 1}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{\pi}{2} \pm \pi\,n\\\frac{\pi}{3} \pm 2\pi\,n\\\\\frac{5\pi}{3} \pm 2\pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    11. $\displaystyle{8\sin^4 x - 10\sin^2 x + 3 = 0}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{\pi}{4} \pm \pi\,n\\\frac{3\pi}{4} \pm \pi\,n\\\\\frac{\pi}{3} \pm \pi\,n\\\\\frac{2\pi}{3} \pm \pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    12. $\displaystyle{2\sin^2 x + 7\sin x + 3 = 0}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{7\pi}{6} \pm 2\pi\,n\\\frac{11\pi}{6} \pm 2\pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$

    13. $\displaystyle{3\cot x = \tan x}$   $\ans{\displaystyle{\left. \begin{array}{ll}\frac{\pi}{3} \pm \pi\,n\\\frac{2\pi}{3} \pm \pi\,n \end{array} \right\} \textrm{ where } n=0,1,2,\ldots}}$