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Mathematics LibreTexts

17.3: Exercises

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Exercise 17.3.1

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

  1. x=120
  2. x=390
  3. x=150
  4. x=45
  5. x=1050
  6. x=810
  7. x=5π4
  8. x=5π6
  9. x=10π3
  10. x=15π2
  11. x=π6
  12. x=54π8
Answer
  1. sin(120)=32,cos(120)=12,tan(120)=3
  2. sin(390)=12,cos(390)=32,tan(390)=33
  3. sin(150)=12,cos(150)=32,tan(150)=33
  4. sin(45)=22,cos(45)=22,tan(45)=1
  5. sin(1050)=12,cos(1050)=32,tan(1050)=33
  6. sin(810)=1,cos(810)=0,tan(810) is undefined
  7. sin(5π4)=22,cos(5π4)=22,tan(5π4)=1
  8. sin(5π6)=12,cos(5π6)=32,tan(5π6)=33
  9. sin(10π3)=32,cos(10π3)=12,tan(10π3)=3
  10. sin(15π2)=1,cos(15π2)=0,tan(15π2) is undefined
  11. sin(π6)=12,cos(π6)=32,tan(π6)=33
  12. sin(54π8)=22,cos(54π8)=22,tan(54π8)=1

Exercise 17.3.2

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(xπ)
  3. f(x)=tan(x)4
  4. f(x)=5sin(x)
  5. f(x)=cos(2x)
  6. f(x)=sin(x2)5
Answer
  1. shift y=sin(x) up by 2 clipboard_e87a9fec74db13512dc88d00c5a408b50.png
  2. y=cos(x) shifted to the right by π 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 17.3.3

Identify the formulas with the graphs. f(x)=sin(x)+2,g(x)=tan(x1),h(x)=3sin(x),i(x)=3cos(x),j(x)=cos(xπ),k(x)=tan(x)1

  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 17.3.4

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=5cos(x)
  2. y=5cos(x)
  3. y=5sin(x)
  4. y=cos(x)+5
  5. y=sin(x)+5
  6. y=2sin(x)+3

Exercise 17.3.5

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

  1. f(x)=5sin(2x+3)
  2. f(x)=sin(πx5)
  3. f(x)=6sin(4x)
  4. f(x)=2cos(x+π4)
  5. f(x)=8cos(2x6)
  6. f(x)=3sin(x4)
  7. f(x)=cos(x+2)
  8. f(x)=7sin(2π5x6π5)
  9. f(x)=cos(2x)
Answer
  1. amplitude 5, period π, phase-shift 32
  2. amplitude 1, period 2, phase-shift 5π
  3. amplitude 6, period π2, phase-shift 0
  4. amplitude 2, period 2π, phase-shift π4
  5. amplitude 8, period π, phase-shift 3
  6. amplitude 3, period 8π, phase-shift 0
  7. amplitude 1, period 2π, phase-shift 2
  8. amplitude 7, period 5, phase-shift 3
  9. amplitude 1, period π, phase-shift 0

Exercise 17.3.6

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=5cos(2x)
  2. y=4sin(πx)
  3. y=2sin(2π3x)
  4. y=cos(2xπ)
  5. y=cos(πxπ)
  6. y=6cos(x4)
  7. y=cos(4x+π)
  8. y=7sin(x+π4)
  9. y=5cos(x+3π2)
  10. y=4sin(5xπ)
  11. y=3cos(2πx4)
  12. y=7sin(14x+π4)
  13. y=cos(3x4π)
  14. y=2sin(15xπ10)
  15. y=13cos(145x6π5)
Answer
  1. amplitude 5, period π, 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 π, phase-shift π2 clipboard_e64231c2bd27d118a37761a9bddfafb9d.png
  5. amplitude 1, period 2, phase-shift 1 clipboard_e7ec6ef7ac4ef9917dd623cbfe2a5ef3d.png
  6. amplitude 6, period 8π, phase-shift 0 clipboard_ed1774e04d9474194870114a705c9b522.png
  7. amplitude 1, period π2, phase-shift π4 clipboard_e3eb28effa7317eab90f2b0de9a48367d.png
  8. amplitude 7, period 2π, phase-shift π4 clipboard_ec0268bfd6521121da67ce54e9eba246b.png
  9. amplitude 5, period 2π, phase-shift 3π2 clipboard_e41db0af7dfe18c464fc5480985a46120.png
  10. amplitude 4, period 2π5, phase-shift π5 clipboard_e1fcc37d96d30151b0cf20a260141868c.png
  11. amplitude 3, period 1, phase-shift 2π clipboard_e6675a6425361cf8d163fd4982cde4056.png
  12. amplitude 7, period 8π, phase-shift π clipboard_ea018243cb0d56c545263af605094776f.png
  13. amplitude 1, period 2π3, phase-shift 4π3 clipboard_e47af696f68a6368c441fb4b5a6f6c833.png
  14. amplitude 2, period 10π, phase-shift π2 clipboard_e32bbf0c4182ae4dc97ce0163f8be391b.png
  15. amplitude 13, period 5π7, phase-shift 3π7 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.

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