Exercises - Trigonometric Functions on the Unit Circle

  1. Find the value of each of the following:  

    1. $\sin \frac{5\pi}{4}$

    2. $\sin \frac{17\pi}{6}$

    3. $\cos \, (-\frac{\pi}{3})$

    4. $\sin \, ( -\frac{5\pi}{2})$

  2. If $\sin t = 1/4$, and the terminal side of an angle of $t$ radians is in quadrant II, find $\cos t$.

  3. If $\cos t = 3/10$, find all possible values of $\sin t$.   $\ans{
    \begin{array}{rcl}
    \cos^2 t + \sin^2 t &=& 1\\
    (\frac{3}{10})^2 + \sin^2 t &=& 1\\
    \frac{9}{100} + \sin^2 t &=& 1\\
    \sin^2 t &=& \frac{91}{100}\\
    \sin t &=& \pm\frac{\sqrt{91}}{10}
    \end{array}}$

  4. Find $\tan t, \cot t, \sec t,$ and $\csc t$ for $t = -\frac{\pi}{3}$.   $\ans{
    \begin{array}{rcl}
    \sin t &=& -\frac{\sqrt{3}}{2} \textrm{ from the unit circle}\\
    \cos t &=& \frac{1}{2} \textrm{ from the unit circle}\\
    \tan t &=& \frac{\sin t}{\cos t} = -\sqrt{3}\\
    \cot t &=& \frac{\cos t}{\sin t} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\\
    \sec t &=& \frac{1}{\cos t} = 2\\
    \csc t &=& \frac{1}{\sin t} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}
    \end{array}}$

  5. Find the value of each expression given

    1. $\sec \, \frac{7\pi}{6}$   $\ans{\cos \frac{7\pi}{6} = -\frac{\sqrt{3}}{2}, \textrm{ so } \sec \frac{7\pi}{6} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}}$

    2. $\csc \, (-\frac{3\pi}{2})$   $\ans{\sin (-\frac{3\pi}{2}) = 1, \textrm{ so } \csc (-\frac{3\pi}{2}) = 1/1 = 1}$

    3. $\cot \, ( -\frac{3\pi}{2})$   $\ans{\cos (-\frac{3\pi}{2}) = 0 \textrm{ and } \sin (-\frac{3\pi}{2}) = 1, \textrm{ so } \cot (-\frac{3\pi}{2}) = 0/1 = 0}$

    4. $\tan \, (\frac{3\pi}{4})$   $\ans{\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \textrm{ and } \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}, \textrm{ so } \tan \frac{3\pi}{4} = \frac{\sqrt{2}/2}{-\sqrt{2}/2} = -1}$

  6. Find $\sin \theta$, if $\cot \theta = \frac{3}{4}$ and $\pi \lt \theta \lt \frac{3\pi}{2}$   $\ans{
    \begin{array}{rcl}
    \cot \theta = \frac{3}{4} &\rightarrow& \frac{\cos \theta}{\sin \theta} = \frac{3}{4}\\\\
    &\rightarrow& \cos \theta = \frac{3}{4} \sin \theta\\\\
    &\rightarrow& 1 = \cos^2 \theta + \sin^2 \theta = \left( \frac{3}{4} \sin \theta \right)^2 + \sin^2 \theta\\\\
    &\rightarrow& 1 = \frac{9}{16} \sin^2 \theta + \sin^2 \theta\\\\
    &\rightarrow& 1 = \frac{25}{16} \sin^2 \theta\\\\
    &\rightarrow& \frac{16}{25} = \sin^2 \theta\\\\
    &\rightarrow& \sin \theta = \pm \frac{4}{5}\\\\
    &\rightarrow& \sin \theta = -\frac{4}{5}, \textrm{ since when } \pi \lt \theta \lt \frac{3\pi}{2}, \sin \theta \textrm{ is negative}
    \end{array}}$

  7. Find $\sec \theta$, if $\csc \theta = -3$ and $\frac{3\pi}{2} \lt \theta \lt 2\pi$   $\ans{
    \begin{array}{rcl}
    \csc \theta = -3 &\rightarrow& \sin \theta = -\frac{1}{3}\\\\
    &\rightarrow& 1 = \cos^2 \theta + \sin^2 \theta = \cos^2 \theta + (-\frac{1}{3})^2\\\\
    &\rightarrow& 1 = \cos^2 \theta + \frac{1}{9}\\\\
    &\rightarrow& \frac{8}{9} = \cos^2 \theta\\\\
    &\rightarrow& \cos \theta = \pm \frac{2\sqrt{2}}{3}\\\\
    &\rightarrow& \cos \theta = \frac{2\sqrt{2}}{3}, \textrm{ since when } \frac{3\pi}{2} \lt \theta \lt 2\pi, \cos \theta \textrm{ is positive}\\\\
    &\rightarrow& \sec \theta = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4}
    \end{array}\\}$

  8. Find the values of the following:

    1. $\cos 5\pi$   $\ans{\displaystyle{-1}}$

    2. $\sin \, (-\frac{7\pi}{6})$   $\ans{\displaystyle{\frac{1}{2}}}$

    3. $\cos \, (\frac{23\pi}{4})$   $\ans{\displaystyle{\frac{\sqrt{2}}{2}}}$

    4. $\sin 9\pi$   $\ans{\displaystyle{0}}$

    5. $\cos \, (-\frac{10\pi}{3})$   $\ans{\displaystyle{-\frac{1}{2}}}$

    6. $\sin \, (-\frac{4\pi}{3})$   $\ans{\displaystyle{\frac{\sqrt{3}}{2}}}$

    7. $\cot \, (\frac{13\pi}{6})$   $\ans{\displaystyle{\sqrt{3}}}$

    8. $\tan \, (\frac{9\pi}{2})$   $\ans{\displaystyle{\textrm{does not exist}}}$

    1. $\csc \, (-\frac{\pi}{6})$   $\ans{\displaystyle{-2}}$

    2. $\tan \, (\frac{23\pi}{4})$   $\ans{\displaystyle{-1}}$

    3. $\sec \, (\frac{10\pi}{3})$   $\ans{\displaystyle{-2}}$

    4. $\csc 5\pi$   $\ans{\displaystyle{\textrm{does not exist}}}$

    5. $\cot \, (-\frac{5\pi}{4})$   $\ans{\displaystyle{-1}}$

    6. $\sin 150^{\circ}$   $\ans{\displaystyle{\frac{1}{2}}}$

    7. $\sec -120^{\circ}$   $\ans{\displaystyle{-2}}$

    8. $\csc 495^{\circ}$   $\ans{\displaystyle{\sqrt{2}}}$


  9. Make a table (the heading and beginning of which are shown below) giving the values of all six trigonometric functions for
    $$t=0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}, \pi, \frac{7\pi}{6}, \frac{5\pi}{4}, \frac{4\pi}{3}, \frac{3\pi}{2}, \frac{5\pi}{3}, \frac{7\pi}{4}, \textrm{ and } \frac{11\pi}{6}$$
    $$\begin{array}{|c|c|c|c|c|c|c|}
    t & \cos t & \sin t & \tan t & \cot t & \sec t & \csc t\\\hline
    0 & 1 & 0 & 0 & - & 1 & - \\
    \pi/6 & \sqrt{3}/2 & 1/2 & & & & \\
    \pi/4 & & & & & &
    \end{array}$$

  10. Find the value of

    1. $\cos t$, if $\tan t = -\frac{2}{3}$ and $\frac{3\pi}{2} \lt t \lt 2\pi$   $\ans{\displaystyle{\frac{3\sqrt{13}}{13}}}$

    2. $\sin t$, if $\sec t = \frac{13}{5}$ and $0 \lt t \lt \frac{\pi}{2}$   $\ans{\displaystyle{\frac{12}{13}}}$

    3. $\tan t$, if $\csc t = \frac{5}{3}$ and $\frac{\pi}{2} \lt t \lt \pi$   $\ans{\displaystyle{-\frac{3}{4}}}$

    4. $\cot t$, if $\csc t = \frac{5}{4}$ and $0 \lt t \lt \frac{\pi}{2}$   $\ans{\displaystyle{\frac{3}{4}}}$

    5. $\sin \theta$, if $\cot \theta = -\frac{4}{9}$ and $\frac{\pi}{2} \lt \theta \lt \pi$   $\ans{\displaystyle{\frac{9\sqrt{97}}{97}}}$

    6. $\cos \theta$, if $\tan \theta = \frac{\sqrt{3}}{2}$ and $\pi \lt \theta \lt \frac{3\pi}{2}$   $\ans{\displaystyle{-\frac{2\sqrt{7}}{7}}}$

    7. $\sec \theta$, if $\sin \theta = -\frac{1}{6}$ and $\frac{3\pi}{2} \lt \theta \lt 2\pi$   $\ans{\displaystyle{\frac{6\sqrt{35}}{35}}}$

    8. $\csc \theta$, if $\cot \theta = -\frac{\sqrt{18}}{12}$ and $\frac{\pi}{2} \lt \theta \lt \pi$   $\ans{\displaystyle{\frac{3\sqrt{2}}{4}}}$