20.3: Exercises
- Page ID
- 49082
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Find all solutions of the equation, and simplify as much as possible. Do not approximate the solution.
- \(\tan(x)=\dfrac{\sqrt{3}}{3}\)
- \(\sin(x)=\dfrac{\sqrt{3}}{2}\)
- \(\sin(x)=-\dfrac{\sqrt{2}}{2}\)
- \(\cos(x)=\dfrac{\sqrt{3}}{2}\)
- \(\cos(x)=0\)
- \(\cos(x)=-0.5\)
- \(\cos(x)=1\)
- \(\sin(x)=5\)
- \(\sin(x)=0\)
- \(\sin(x)=-1\)
- \(\tan(x)=-\sqrt{3}\)
- \(\cos(x)=0.2\)
- Answer
-
- \(x=\dfrac{\pi}{6}+n \pi \text {, where } n=0, \pm 1, \ldots\)
- \(x=(-1)^{n} \dfrac{\pi}{3}+n \pi \text {, where } n=0, \pm 1, \ldots\)
- \(x=(-1)^{n+1} \dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{\pi}{2}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{2 \pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- there is no solution (since \(-1 \leq \sin (x) \leq\)),
- \(x=n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots\) (since each solution appears twice, it is enough to take \(n=0, \pm 2, \pm 4, \ldots\)),
- \(x=\dfrac{-\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \cos ^{-1}(0.2)+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
Find all solutions of the equation. Approximate your solution with the calculator.
- \(\tan(x)=6.2\)
- \(\cos(x)=0.45\)
- \(\sin(x)=0.91\)
- \(\cos(x)=-.772\)
- \(\tan(x)=-0.2\)
- \(\sin(x)=-0.06\)
- Answer
-
- \(x \approx 1.411+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx \pm 1.104+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx(-1)^{n} 1.143+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx \pm 2.453+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx-0.197+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx(-1)^{n+1} 0.06+n \pi, \text { where } n=0, \pm 1, \ldots\)
Find at least \(5\) distinct solutions of the equation.
- \(\tan(x)=-1\)
- \(\cos(x)=\dfrac{\sqrt{2}}{2}\)
- \(\sin(x)=-\dfrac{\sqrt{3}}{2}\)
- \(\tan(x)=0\)
- \(\cos(x)=0\)
- \(\cos(x)=0.3\)
- \(\sin(x)=0.4\)
- \(\sin(x)=-1\)
- Answer
-
- \(\dfrac{-\pi}{4}, \dfrac{3 \pi}{4}, \dfrac{7 \pi}{4}, \dfrac{-5 \pi}{4}, \dfrac{-9 \pi}{4}\)
- \(\dfrac{\pi}{4}, \dfrac{-\pi}{4}, \dfrac{9 \pi}{4}, \dfrac{-9 \pi}{4}, \dfrac{17 \pi}{4}, \dfrac{-17 \pi}{4}\)
- \(\dfrac{-\pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}, \dfrac{-2 \pi}{3}, \dfrac{-7 \pi}{3}\)
- \(0, \pi, 2 \pi,-\pi,-2 \pi,\)
- \(\dfrac{\pi}{2}, \dfrac{-\pi}{2}, \dfrac{3 \pi}{2}, \dfrac{-3 \pi}{2}, \dfrac{5 \pi}{2}, \dfrac{-5 \pi}{2}\)
- \(\cos ^{-1}(0.3),-\cos ^{-1}(0.3), \cos ^{-1}(0.3)+2 \pi,-\cos ^{-1}(0.3)+2 \pi, \cos ^{-1}(0.3)-2 \pi,-\cos ^{-1}(0.3)-2 \pi\)
- \(\sin ^{-1}(0.4),-\sin ^{-1}(0.4)+\pi, -\sin ^{-1}(0.4)-\pi, \sin ^{-1}(0.4)+2 \pi, \sin ^{-1}(0.4)-2 \pi\)
- \(\dfrac{3 \pi}{2}, \dfrac{7 \pi}{2}, \dfrac{11 \pi}{2}, \dfrac{-\pi}{2}, \dfrac{-5 \pi}{2}\)
Solve for \(x\). State the general solution without approximation.
- \(\tan(x)-1=0\)
- \(2\sin(x)=1\)
- \(2\cos(x)-\sqrt{3}=0\)
- \(\sec(x)=-2\)
- \(\cot(x)=\sqrt{3}\)
- \(\tan^2(x)-3=0\)
- \(\sin^2(x)-1=0\)
- \(\cos^2(x)+7\cos(x)+6=0\)
- \(4\cos^2(x)-4\cos(x)+1=0\)
- \(2\sin^2(x)+11\sin(x)=-5\)
- \(2\sin^2(x)+\sin(x)-1=0\)
- \(2\cos^2(x)-3\cos(x)+1=0\)
- \(2\cos^2(x)+9\cos(x)=5\)
- \(\tan^3(x)-\tan(x)=0\)
- Answer
-
- \(x=\dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=(-1)^{n} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{2 \pi}{3}+2 n \pi \text { where } n=0, \pm 1, \ldots\)
- \(x=\dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pi+2 n \pi \text {, where } n=0, \pm 1, \ldots\) (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 \(x=\pi+2 n \pi\).),
- \(x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=(-1)^{n+1} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi\)
- \(x=2 n \pi, \text { or } x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x=\pm \dfrac{\pi}{4}+n \pi, \text { or } x=n \pi, \text { where } n=0, \pm 1, \ldots\)
Use the calculator to find all solutions of the given equation. Approximate the answer to the nearest thousandth.
- \(2\cos(x)=2\sin(x)+1\)
- \(7\tan(x)\cdot \cos(2x)=1\)
- \(4\cos^2(3x)+\cos(3x)=\sin(3x)+2\)
- \(\sin(x)+\tan(x)=\cos(x)\)
- Answer
-
- \(x \approx-1.995+2 n \pi, \text { or } x \approx 0.424+2 n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx-0.848+n \pi, \text { or } x \approx 0.148+n \pi, \text { or } x \approx 0.700+n \pi, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx 0.262+n \dfrac{2 \pi}{3} \text {, or } x \approx 0.906+n \dfrac{2 \pi}{3} \text {, or } x \approx 1.309+n \dfrac{2 \pi}{3}, \text { or } x \approx 1.712+n \dfrac{2 \pi}{3}, \text { where } n=0, \pm 1, \ldots\)
- \(x \approx 0.443+2 n \pi, \text { or } x \approx 2.193+2 n \pi, \text { where } n=0, \pm 1, \ldots\)