Verifying a Trigonometric Identity...4

Discussion in 'Geometry and Trigonometry' started by nycmathguy, Jan 5, 2022.

  1. nycmathguy

    nycmathguy

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    Section 5.5

    Can you please get me started with 69?

    The plus/minus on the numerator throws me into a loop.

    Screenshot_20220103-223739_Samsung Notes.jpg
     
    nycmathguy, Jan 5, 2022
    #1
  2. nycmathguy

    MathLover1

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    69.

    (sin(x)+-sin(y})/(cos(x)+cos(y))=tan((x+-y)/2)

    separate in two:

    (sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)
    and
    (sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)

    prove one:

    (sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)

    manipulate left side

    (sin(x)+sin(y))/(cos(x)+cos(y)).....rewrite using trigonometric identities

    sin(x)+sin(y)=2sin((x+y)/2)cos((x-y)/2)

    cos(x)+cos(y)=2cos((x+y)/2)cos((x−y)/2)

    then you have

    (sin(x)+sin(y))/(cos(x)+cos(y))

    =(2sin((x+y)/2)cos((x-y)/2))/2cos((x+y)/2)cos((x−y)/2)........simplify

    =sin((x+y)/2))/cos((x+y)/2)...........use identity tan(x)=sin(x)/cos(x)

    =tan((x+y)/2))

    same is for (sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)
     
    MathLover1, Jan 5, 2022
    #2
    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

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    I was going to work it out but I thank you.
    Go check out what Country Boy been saying about your replies to my threads in the Algebra forum. In correcting your replies, he has confused me even more.

    One more for today:

    Trigonometric Equations
     
    nycmathguy, Jan 6, 2022
    #3
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