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17.3: Exercises

  • Page ID
    49061
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    Exercise \(\PageIndex{1}\)

    Find \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\) for the following angles.

    1. \(x=120^\circ\)
    2. \(x=390^\circ\)
    3. \(x=-150^\circ\)
    4. \(x=-45^\circ\)
    5. \(x=1050^\circ\)
    6. \(x=-810^\circ\)
    7. \(x=\dfrac{5\pi}{4}\)
    8. \(x=\dfrac{5\pi}{6}\)
    9. \(x=\dfrac{10\pi}{3}\)
    10. \(x=\dfrac{15\pi}{2}\)
    11. \(x=\dfrac{-\pi}{6}\)
    12. \(x=\dfrac{-54\pi}{8}\)
    Answer
    1. \(\sin \left(120^{\circ}\right)=\dfrac{\sqrt{3}}{2}, \cos \left(120^{\circ}\right)=-\dfrac{1}{2}, \tan \left(120^{\circ}\right)=-\sqrt{3}\)
    2. \(\sin \left(390^{\circ}\right)=\dfrac{1}{2}, \cos \left(390^{\circ}\right)=\dfrac{\sqrt{3}}{2}, \tan \left(390^{\circ}\right)=\dfrac{\sqrt{3}}{3}\)
    3. \(\sin \left(-150^{\circ}\right)=-\dfrac{1}{2}, \cos \left(-150^{\circ}\right)=-\dfrac{\sqrt{3}}{2}, \tan \left(-150^{\circ}\right)=\dfrac{\sqrt{3}}{3}\)
    4. \(\sin \left(-45^{\circ}\right)=-\dfrac{\sqrt{2}}{2}, \cos \left(-45^{\circ}\right)=\dfrac{\sqrt{2}}{2}, \tan \left(-45^{\circ}\right)=-1\)
    5. \(\sin \left(1050^{\circ}\right)=-\dfrac{1}{2}, \cos \left(1050^{\circ}\right)=\dfrac{\sqrt{3}}{2}, \tan \left(1050^{\circ}\right)=-\dfrac{\sqrt{3}}{3}\)
    6. \(\sin \left(-810^{\circ}\right)=-1, \cos \left(-810^{\circ}\right)=0, \tan \left(-810^{\circ}\right)\) is undefined
    7. \(\sin \left(\dfrac{5 \pi}{4}\right)=-\dfrac{\sqrt{2}}{2}, \cos \left(\dfrac{5 \pi}{4}\right)=-\dfrac{\sqrt{2}}{2}, \tan \left(\dfrac{5 \pi}{4}\right)=1\)
    8. \(\sin \left(\dfrac{5 \pi}{6}\right)=\dfrac{1}{2}, \cos \left(\dfrac{5 \pi}{6}\right)=-\dfrac{\sqrt{3}}{2}, \tan \left(\dfrac{5 \pi}{6}\right)=-\dfrac{\sqrt{3}}{3}\)
    9. \(\sin \left(\dfrac{10 \pi}{3}\right)=-\dfrac{\sqrt{3}}{2}, \cos \left(\dfrac{10 \pi}{3}\right)=-\dfrac{1}{2}, \tan \left(\dfrac{10 \pi}{3}\right)=\sqrt{3}\)
    10. \(\sin \left(\dfrac{15 \pi}{2}\right)=-1, \cos \left(\dfrac{15 \pi}{2}\right)=0, \tan \left(\dfrac{15 \pi}{2}\right)\) is undefined
    11. \(\sin \left(\dfrac{-\pi}{6}\right)=-\dfrac{1}{2}, \cos \left(\dfrac{-\pi}{6}\right)=\dfrac{\sqrt{3}}{2}, \tan \left(\dfrac{-\pi}{6}\right)=-\dfrac{\sqrt{3}}{3}\)
    12. \(\sin \left(\dfrac{-54 \pi}{8}\right)=-\dfrac{\sqrt{2}}{2}, \cos \left(\dfrac{-54 \pi}{8}\right)=-\dfrac{\sqrt{2}}{2}, \tan \left(\dfrac{-54 \pi}{8}\right)=1\)

    Exercise \(\PageIndex{2}\)

    Graph the function, and describe how the graph can be obtained from one of the basic graphs \(y=\sin(x)\), \(y=\cos(x)\), or \(y=\tan(x)\).

    1. \(f(x)=\sin(x)+2\)
    2. \(f(x)=\cos(x-\pi)\)
    3. \(f(x)=\tan(x)-4\)
    4. \(f(x)=5\cdot \sin(x)\)
    5. \(f(x)=\cos(2\cdot x)\)
    6. \(f(x)=\sin(x-2)-5\)
    Answer
    1. shift \(y = \sin(x)\) up by \(2\) clipboard_e87a9fec74db13512dc88d00c5a408b50.png
    2. \(y = \cos(x)\) shifted to the right by \(\pi\) clipboard_ea49e58b7b0e659038bf9635b9ba96f8e.png
    3. \(y = \tan(x)\) shifted down by \(4\) clipboard_e4568fbe75b058d26e053e360fb41df95.png
    4. \(y = \sin(x)\) stretched away from the \(x\)-axis by a factor \(5\) clipboard_ea70bf9f705034f50ed58be4f251a6281.png
    5. \(y = \cos(x)\) compressed towards the \(y\)-axis by a factor \(2\) clipboard_e080744b55078ea167a3afba862c78862.png
    6. \(y = \sin(x)\) shifted to the right by \(2\) and down by \(5\) clipboard_e7b1355ee1b59ad066326d55ba2459d2a.png

    Exercise \(\PageIndex{3}\)

    Identify the formulas with the graphs. \[\begin{array}{lll}
    f(x)=\sin (x)+2, & g(x)=\tan (x-1), & h(x)=3 \sin (x), \\
    i(x)=3 \cos (x), & j(x)=\cos (x-\pi), & k(x)=\tan (x)-1
    \end{array} \nonumber \]

    1. clipboard_ed9f7f6deab20c9a085dbd10a02f684dd.png
    2. clipboard_e020683b23064247032aac69fc1c7a107.png
    3. clipboard_e5469166b94ce909f729ca288a1aed298.png
    4. clipboard_e06decdf4a038417807f651176e3dd831.png
    5. clipboard_e51f2385d56e4e68d886ca212744a7316.png
    6. clipboard_ecf741760cde398a86b7a8ec8776b2587.png
    Answer
    1. \(g(x)\)
    2. \(h(x)\)
    3. \(j(x)\)
    4. \(k(x)\)
    5. \(i(x)\)
    6. \(f(x)\)

    Exercise \(\PageIndex{4}\)

    Find the formula of a function whose graph is the one displayed below.

    1. clipboard_ea430c8007e6555ae5fafacf64920c0b9.png
    2. clipboard_eef7dfc0304d8751afc660afd06d91a26.png
    3. clipboard_e3cdb414f98f2d1e143a135131b7718ba.png
    4. clipboard_ea748c6feb830be4ec4d0323292225119.png
    5. clipboard_ec442cb9f2d0e69954a33c055884fafe1.png
    6. clipboard_ecf5d99db8f8e0d021c4d07ffcfef9c6e.png
    Answer
    1. \(y = 5 \cos(x)\)
    2. \(y = −5 \cos(x)\)
    3. \(y = −5 \sin(x)\)
    4. \(y = \cos(x) + 5\)
    5. \(y = \sin(x) + 5\)
    6. \(y = 2 \sin(x) + 3\)

    Exercise \(\PageIndex{5}\)

    Find the amplitude, period, and phase-shift of the function.

    1. \(f(x)=5\sin(2x+3)\)
    2. \(f(x)=\sin(\pi x-5)\)
    3. \(f(x)=6\sin(4x)\)
    4. \(f(x)=-2\cos\left(x+\dfrac{\pi}{4}\right)\)
    5. \(f(x)=8\cos(2x-6)\)
    6. \(f(x)=3\sin\left(\dfrac{x}{4}\right)\)
    7. \(f(x)=-\cos(x+2)\)
    8. \(f(x)=7\sin \left(\dfrac{2\pi}{5}x-\dfrac{6\pi}{5}\right)\)
    9. \(f(x)=\cos(-2x)\)
    Answer
    1. amplitude \(5\), period \(\pi\), phase-shift \(\dfrac{−3}{2}\)
    2. amplitude \(1\), period \(2\), phase-shift \(\dfrac 5 \pi\)
    3. amplitude \(6\), period \(\dfrac \pi 2\), phase-shift \(0\)
    4. amplitude \(2\), period \(\dfrac 2 \pi\), phase-shift \(\dfrac{−\pi}{4}\)
    5. amplitude \(8\), period \(\pi\), phase-shift \(3\)
    6. amplitude \(3\), period \(\dfrac 8 \pi\), phase-shift \(0\)
    7. amplitude \(1\), period \(\dfrac 2 \pi\), phase-shift \(−2\)
    8. amplitude \(7\), period \(5\), phase-shift \(3\)
    9. amplitude \(1\), period \(\pi\), phase-shift \(0\)

    Exercise \(\PageIndex{6}\)

    Find the amplitude, period, and phase-shift of the function. Use this information to graph the function over a full period. Label all maxima, minima, and zeros of the function.

    1. \(y=5\cos(2x)\)
    2. \(y=4\sin(\pi x)\)
    3. \(y=2\sin\left(\dfrac{2\pi}{3}x\right)\)
    4. \(y=\cos(2x-\pi)\)
    5. \(y=\cos(\pi x-\pi)\)
    6. \(y=-6\cos(-\dfrac{x}{4})\)
    7. \(y=-\cos(4x+\pi)\)
    8. \(y=7\sin\left(x+\dfrac{\pi}{4}\right)\)
    9. \(y=5\cos\left(x+\dfrac{3\pi}{2}\right)\)
    10. \(y=4\sin(5x-\pi)\)
    11. \(y=-3\cos(2\pi x-4)\)
    12. \(y=7\sin\left(\dfrac 1 4 x+\dfrac{\pi}{4}\right)\)
    13. \(y=\cos(3x-4\pi)\)
    14. \(y=2\sin\big(\dfrac 1 5 x-\dfrac{\pi}{10}\big)\)
    15. \(y=\dfrac 1 3 \cos\left(\dfrac{14}{5}x-\dfrac{6\pi}{5}\right)\)
    Answer
    1. amplitude \(5\), period \(\pi\), phase-shift \(0\) clipboard_e113627ff02d07cc9bea67038b4d8a974.png
    2. amplitude \(4\), period \(2\), phase-shift \(0\) clipboard_edef39b269c155473be986ec93e2577c2.png
    3. amplitude \(2\), period \(3\), phase-shift \(0\) clipboard_e6ab47a8cab9e849a888344707a04ecda.png
    4. amplitude \(1\), period \(\pi\), phase-shift \(\dfrac \pi 2\) clipboard_e64231c2bd27d118a37761a9bddfafb9d.png
    5. amplitude \(1\), period \(2\), phase-shift \(1\) clipboard_e7ec6ef7ac4ef9917dd623cbfe2a5ef3d.png
    6. amplitude \(6\), period \(\dfrac 8 \pi \), phase-shift \(0\) clipboard_ed1774e04d9474194870114a705c9b522.png
    7. amplitude \(1\), period \(\dfrac \pi 2\), phase-shift \(\dfrac {−\pi}{4}\) clipboard_e3eb28effa7317eab90f2b0de9a48367d.png
    8. amplitude \(7\), period \(\dfrac 2 \pi\), phase-shift \(\dfrac {−\pi}{4}\) clipboard_ec0268bfd6521121da67ce54e9eba246b.png
    9. amplitude \(5\), period \(\dfrac 2 \pi\), phase-shift \(\dfrac {−3\pi}{2}\) clipboard_e41db0af7dfe18c464fc5480985a46120.png
    10. amplitude \(4\), period \(\dfrac {2\pi}{5}\), phase-shift \(\dfrac \pi 5\) clipboard_e1fcc37d96d30151b0cf20a260141868c.png
    11. amplitude \(3\), period \(1\), phase-shift \(\dfrac 2 \pi\) clipboard_e6675a6425361cf8d163fd4982cde4056.png
    12. amplitude \(7\), period \(\dfrac 8 \pi\), phase-shift \(-\pi\) clipboard_ea018243cb0d56c545263af605094776f.png
    13. amplitude \(1\), period \(\dfrac {2\pi}{3}\), phase-shift \(\dfrac {4\pi}{3}\) clipboard_e47af696f68a6368c441fb4b5a6f6c833.png
    14. amplitude \(2\), period \(\dfrac 10 \pi\), phase-shift \(\dfrac \pi 2\) clipboard_e32bbf0c4182ae4dc97ce0163f8be391b.png
    15. amplitude \(\dfrac 1 3\), period \(\dfrac {5\pi}{7}\), phase-shift \(\dfrac {3\pi}{7}\) clipboard_ef80d92a6baf65f7530888a753667da1c.png

    This page titled 17.3: 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.