Trigonometric Equations

Discussion in 'Geometry and Trigonometry' started by nycmathguy, Dec 26, 2021.

  1. nycmathguy

    nycmathguy

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    Section 5.4

    Can you help me with 69 and 71?

    Screenshot_20211226-094501_Samsung Notes.jpg
     

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    nycmathguy, Dec 26, 2021
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  2. nycmathguy

    MathLover1

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    69.

    sin(x+pi)-sin(x)+1=0........... [0,2π)

    sin(x+pi)=-sin(x)

    -sin(x)-sin(x)+1=0

    1-2sin(x)=0
    1=2sin(x)
    sin(x) =1/2
    x =sin^-1(1/2)
    x=π/6
    general solutions:

    x=π/6+2π*n

    solutions in interval [0,2π):

    Radians:
    x=π/6
    x=5πi/6

    Degrees:
    x=30 °
    x=150 °



    71.

    cos(x+π/4)-cos(x-π/4)=1 .......... [0,2π)

    =>cos(x+π/4)=cos(x)/sqrt(2) - sin(x)/sqrt(2)
    => cos(x-π/4)=sin(x)/sqrt(2) + cos(x)/sqrt(2)

    then
    cos(x)/sqrt(2) - sin(x)/sqrt(2) -(sin(x)/sqrt(2) + cos(x)/sqrt(2))=1
    cos(x)/sqrt(2) - sin(x)/sqrt(2) -sin(x)/sqrt(2) - cos(x)/sqrt(2))=1.......simplify
    -sqrt(2) sin(x)=1
    sin(x)=1/-sqrt(2)
    x=sin^-1(1/-sqrt(2))
    x=-π/4

    general solutions:
    x=-π/4+2pi*n
    Solutions for\ the range : 0<=x<2pi

    n=1
    x=-π/4+2pi
    x = (7π)/4->315° (degrees)

    no solution for x in given interval [0,2π)
     
    MathLover1, Dec 27, 2021
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    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

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    Thank you. I will do 2 or 3 additional trig equations when time allows. As you know, my weekend is over.
     
    nycmathguy, Dec 27, 2021
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