Exercises - Trigonometric Equations

  1. Solve the following.

    1.   $\displaystyle{2\sin x = 1}$  

    2.   $\displaystyle{4\cos^2 x - 8 \cos x + 3 = 0}$  

    3.   $\displaystyle{2\sin^2 x + (2- \sqrt{3}) \sin x - \sqrt{3} = 0}$  

    4.   $\displaystyle{\sin x = \cos x}$  

    5.   $\displaystyle{3\cos^2 \theta - \cos 2\theta = 1}$  

    6.   $\displaystyle{\sin x + \sqrt{\sin x} = 0}$  

    7.   $\displaystyle{\sec x \sin^2 x = \tan x}$  

    8.   $\displaystyle{\cos x \sqrt{1 + \tan^2 x} = 1}$  

  2. Find all solutions for the following equations.

    1.   $\displaystyle{\tan x = 0}$   $\ans{\displaystyle{0 \pm \pi \,n \textrm{ where } n=0,1,2,\ldots}}$

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

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

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

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

    6.   $\displaystyle{3\sec^2 x = \sec x}$   $\ans{\displaystyle{\textrm{no solutions}}}$

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

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

    9.   $\displaystyle{\frac{1+\cos x}{\cos x} = 2}$   $\ans{\displaystyle{0 \pm 2\pi\,n \textrm{ where } n=0,1,2,\ldots}}$

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