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Simplifying trig expression

  1. Jun 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Simplify and write the trigonometric expression in terms of sine and cosine:

    (sec (t) - cos (t))/sec (t) = (f(t))^2

    2. Relevant equations

    sec (t)=1/cos (t)

    3. The attempt at a solution

    (sec (t) - cos (t))/sec (t)

    = ((1/cos (t))-cos(t)) / (1/cos (t))

    = ((1-cos^2(t))/cos(t)) / (1/cos (t))

    From here, can I take the entire numerator, ((1-cos^2(t))/cos(t)), and divide it by one? This way I can do division of two rational numbers to get:

    = cos(t)*((1-cos^2(t))/cos(t))

    =1-cos^2(t) = sin^2(t) = (f(t))^2

    so f(t)=sin (t)

    I am quite sure this is the right answer, but I am wondering if the method I used is correct math. Thanks for your help everyone.
     
  2. jcsd
  3. Jun 24, 2008 #2
    That looks correct and your method is spot on.
     
  4. Jun 24, 2008 #3

    dlgoff

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    Gold Member

    = ((1/cos (t))-cos(t)) / (1/cos (t)) goes directly to 1-cos^2(t) by inverting the denominator and multiplying.

    Save a step or two.
     
  5. Jun 24, 2008 #4
    Thanks guys. I have another question as well, this one is as follows:

    1. The problem statement, all variables and given/known data

    sin(x)tan(x) = A) tan (x)
    B) cos (x)
    C) (1-cos^2(x)) / cos(x)

    2. Relevant equations

    tan (x) = sin(x)/cos(x)

    3. The attempt at a solution

    sin(x)tan(x)

    =sin(x) * (sin(x)/cos(x))

    =sin^2(x) / cos(x)

    This is as far as I could simplify, and I can't see how it equals either a, b, or c. Did I make a mistake or is there a step I'm not seeing? Thanks again for your help.
     
  6. Jun 24, 2008 #5

    rock.freak667

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    Homework Helper

    Know any useful trig identities with sin^2(x) and 1 in it?
     
  7. Jun 24, 2008 #6
    haha yea as soon as I posted this I remembered good 'ol Pythagoras
     
  8. Jun 25, 2008 #7
    alright, I have one more question.

    1. The problem statement, all variables and given/known data

    By using known trig identities, sin(2x)/(1+cos(2x)) can be written as:

    A) tan(2x)
    B) tan(x)
    C) csc(2x)
    D) sec(x)
    E) all of the above
    F) none of the above

    2. Relevant equations

    cos x = sin x/cos x

    3. The attempt at a solution

    sin (2x)/ (1+cos(2x))

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

    =sin (2x)+tan (2x)

    This is all I have gotten, and don't feel like getting tan is probably going to help. I also proved that sin x=1+cos x but I haven't gotten anywhere with that either.

    Thanks for any help!
     
  9. Jun 25, 2008 #8

    rock.freak667

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    Homework Helper

    Try your double angle formulas for sin2x and cos2x
     
  10. Jun 25, 2008 #9
    Alright, so:

    2. Relevant equations

    sin (2x) = 2sin(x)cos(x)

    cos (2x) = cos^2(x)-sin^2(x)

    3. The attempt at a solution

    Sin (2x) / (1+cos(2x))

    = (2sin(x)cos(x)) / (1+cos^2(x)-sin^2(x))

    = 2 / (1+cos(x)-sin(x))

    = 2sec(x)-csc(x)

    = sec(2x)-csc(x)

    This seems right, but it doesn't fit with any of the given answers? Did I trip up somewhere along the way?
     
  11. Jun 25, 2008 #10

    rock.freak667

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    Homework Helper

    the parts in red are incorrect.

    From this line


    [tex]\frac{2sinxcosx}{1+cos^2x-sin^2x}[/tex]


    use [itex]sin^2x+cos^2x=1[/itex] and you'll get through.
     
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