Skip to main content
Mathematics LibreTexts

20.3: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    49082

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Exercise \(\PageIndex{1}\)

    Find all solutions of the equation, and simplify as much as possible. Do not approximate the solution.

    1. \(\tan(x)=\dfrac{\sqrt{3}}{3}\)
    2. \(\sin(x)=\dfrac{\sqrt{3}}{2}\)
    3. \(\sin(x)=-\dfrac{\sqrt{2}}{2}\)
    4. \(\cos(x)=\dfrac{\sqrt{3}}{2}\)
    5. \(\cos(x)=0\)
    6. \(\cos(x)=-0.5\)
    7. \(\cos(x)=1\)
    8. \(\sin(x)=5\)
    9. \(\sin(x)=0\)
    10. \(\sin(x)=-1\)
    11. \(\tan(x)=-\sqrt{3}\)
    12. \(\cos(x)=0.2\)
    Answer
    1. \(x=\dfrac{\pi}{6}+n \pi \text {, where } n=0, \pm 1, \ldots\)
    2. \(x=(-1)^{n} \dfrac{\pi}{3}+n \pi \text {, where } n=0, \pm 1, \ldots\)
    3. \(x=(-1)^{n+1} \dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    4. \(x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    5. \(x=\pm \dfrac{\pi}{2}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    6. \(x=\pm \dfrac{2 \pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    7. \(x=2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    8. there is no solution (since \(-1 \leq \sin (x) \leq\)),
    9. \(x=n \pi, \text { where } n=0, \pm 1, \ldots\)
    10. \(x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots\) (since each solution appears twice, it is enough to take \(n=0, \pm 2, \pm 4, \ldots\)),
    11. \(x=\dfrac{-\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    12. \(x=\pm \cos ^{-1}(0.2)+2 n \pi, \text { where } n=0, \pm 1, \ldots\)

    Exercise \(\PageIndex{2}\)

    Find all solutions of the equation. Approximate your solution with the calculator.

    1. \(\tan(x)=6.2\)
    2. \(\cos(x)=0.45\)
    3. \(\sin(x)=0.91\)
    4. \(\cos(x)=-.772\)
    5. \(\tan(x)=-0.2\)
    6. \(\sin(x)=-0.06\)
    Answer
    1. \(x \approx 1.411+n \pi, \text { where } n=0, \pm 1, \ldots\)
    2. \(x \approx \pm 1.104+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    3. \(x \approx(-1)^{n} 1.143+n \pi, \text { where } n=0, \pm 1, \ldots\)
    4. \(x \approx \pm 2.453+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    5. \(x \approx-0.197+n \pi, \text { where } n=0, \pm 1, \ldots\)
    6. \(x \approx(-1)^{n+1} 0.06+n \pi, \text { where } n=0, \pm 1, \ldots\)

    Exercise \(\PageIndex{3}\)

    Find at least \(5\) distinct solutions of the equation.

    1. \(\tan(x)=-1\)
    2. \(\cos(x)=\dfrac{\sqrt{2}}{2}\)
    3. \(\sin(x)=-\dfrac{\sqrt{3}}{2}\)
    4. \(\tan(x)=0\)
    5. \(\cos(x)=0\)
    6. \(\cos(x)=0.3\)
    7. \(\sin(x)=0.4\)
    8. \(\sin(x)=-1\)
    Answer
    1. \(\dfrac{-\pi}{4}, \dfrac{3 \pi}{4}, \dfrac{7 \pi}{4}, \dfrac{-5 \pi}{4}, \dfrac{-9 \pi}{4}\)
    2. \(\dfrac{\pi}{4}, \dfrac{-\pi}{4}, \dfrac{9 \pi}{4}, \dfrac{-9 \pi}{4}, \dfrac{17 \pi}{4}, \dfrac{-17 \pi}{4}\)
    3. \(\dfrac{-\pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}, \dfrac{-2 \pi}{3}, \dfrac{-7 \pi}{3}\)
    4. \(0, \pi, 2 \pi,-\pi,-2 \pi,\)
    5. \(\dfrac{\pi}{2}, \dfrac{-\pi}{2}, \dfrac{3 \pi}{2}, \dfrac{-3 \pi}{2}, \dfrac{5 \pi}{2}, \dfrac{-5 \pi}{2}\)
    6. \(\cos ^{-1}(0.3),-\cos ^{-1}(0.3), \cos ^{-1}(0.3)+2 \pi,-\cos ^{-1}(0.3)+2 \pi, \cos ^{-1}(0.3)-2 \pi,-\cos ^{-1}(0.3)-2 \pi\)
    7. \(\sin ^{-1}(0.4),-\sin ^{-1}(0.4)+\pi, -\sin ^{-1}(0.4)-\pi, \sin ^{-1}(0.4)+2 \pi, \sin ^{-1}(0.4)-2 \pi\)
    8. \(\dfrac{3 \pi}{2}, \dfrac{7 \pi}{2}, \dfrac{11 \pi}{2}, \dfrac{-\pi}{2}, \dfrac{-5 \pi}{2}\)

    Exercise \(\PageIndex{4}\)

    Solve for \(x\). State the general solution without approximation.

    1. \(\tan(x)-1=0\)
    2. \(2\sin(x)=1\)
    3. \(2\cos(x)-\sqrt{3}=0\)
    4. \(\sec(x)=-2\)
    5. \(\cot(x)=\sqrt{3}\)
    6. \(\tan^2(x)-3=0\)
    7. \(\sin^2(x)-1=0\)
    8. \(\cos^2(x)+7\cos(x)+6=0\)
    9. \(4\cos^2(x)-4\cos(x)+1=0\)
    10. \(2\sin^2(x)+11\sin(x)=-5\)
    11. \(2\sin^2(x)+\sin(x)-1=0\)
    12. \(2\cos^2(x)-3\cos(x)+1=0\)
    13. \(2\cos^2(x)+9\cos(x)=5\)
    14. \(\tan^3(x)-\tan(x)=0\)
    Answer
    1. \(x=\dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    2. \(x=(-1)^{n} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    3. \(x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    4. \(x=\pm \dfrac{2 \pi}{3}+2 n \pi \text { where } n=0, \pm 1, \ldots\)
    5. \(x=\dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    6. \(x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    7. \(x=\pm \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    8. \(x=\pi+2 n \pi \text {, where } n=0, \pm 1, \ldots\) (Note: The solution given by the formula 20.1.5 is \(x=\pm \pi+2 n \pi \text { with } n=0, \pm 1, \ldots\) Since every solution appears twice in this expression, we can reduce this to \(x=\pi+2 n \pi\).),
    9. \(x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    10. \(x=(-1)^{n+1} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    11. \(x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi\)
    12. \(x=2 n \pi, \text { or } x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    13. \(x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots\)
    14. \(x=\pm \dfrac{\pi}{4}+n \pi, \text { or } x=n \pi, \text { where } n=0, \pm 1, \ldots\)

    Exercise \(\PageIndex{5}\)

    Use the calculator to find all solutions of the given equation. Approximate the answer to the nearest thousandth.

    1. \(2\cos(x)=2\sin(x)+1\)
    2. \(7\tan(x)\cdot \cos(2x)=1\)
    3. \(4\cos^2(3x)+\cos(3x)=\sin(3x)+2\)
    4. \(\sin(x)+\tan(x)=\cos(x)\)
    Answer
    1. \(x \approx-1.995+2 n \pi, \text { or } x \approx 0.424+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
    2. \(x \approx-0.848+n \pi, \text { or } x \approx 0.148+n \pi, \text { or } x \approx 0.700+n \pi, \text { where } n=0, \pm 1, \ldots\)
    3. \(x \approx 0.262+n \dfrac{2 \pi}{3} \text {, or } x \approx 0.906+n \dfrac{2 \pi}{3} \text {, or } x \approx 1.309+n \dfrac{2 \pi}{3}, \text { or } x \approx 1.712+n \dfrac{2 \pi}{3}, \text { where } n=0, \pm 1, \ldots\)
    4. \(x \approx 0.443+2 n \pi, \text { or } x \approx 2.193+2 n \pi, \text { where } n=0, \pm 1, \ldots\)

    This page titled 20.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.