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Math Help - Trig identitys

  1. #1
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    Trig identitys

    My book lacks answers and i suffer much confusion in this. So here goes, but first quick question. How do i produce signs like pi or theta on here like some of the experts do, along with representing fractions as a fraction as opposed to for example 2x/4

    I misspelled Identities also in title im aware of this :P

    anyway

    tan x csc x cos x =1

    tan=sin x/cos x
    csc x= ? says something like csc(-x)=-csc x
    cos x= 1/sec x

    so i get

    (sin x/cos x)(csc x)(1/sec x)=1 and am stuck here, the csc identity confuses me
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  2. #2
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    Quote Originally Posted by Kromletch View Post
    tan x csc x cos x =1

    tan=sin x/cos x
    Looks like a good start

    \displaystyle \tan x \csc x \cos x

    \displaystyle \frac{\sin x}{\cos x}  \csc x \cos x

    \displaystyle \sin x  \csc x

    (\displaystyle   \csc x  =\frac{1}{\sin x})

    ...
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  3. #3
    MHF Contributor harish21's Avatar
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    To type symbols and fractions and other notations, learn Latex
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    ok what about (4sin^2 x-1)/(2sin x+1)=2sin x-1

    would it be

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

    this is where i get lost again as im not sure how to work the problem. seems like i could cancel out the 2sin x-1 from each side but it seems illegal
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Kromletch View Post
    ok what about (4sin^2 x-1)/(2sin x+1)=2sin x-1

    would it be

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

    this is where i get lost again as im not sure how to work the problem. seems like i could cancel out the 2sin x-1 from each side but it seems illegal
    First, I'm not sure where you got the x + 1 in the denominator.

    Typically you want to work on only one side of the (suspected) identity. In this case
    \displaystyle \frac{4~sin^2(x) - 1}{2~sin(x) + 1}

    Note that the numerator is in the form \displaystyle a^2 - b^2 = (a - b)(a + b). Can you take it from here?

    -Dan
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  6. #6
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    Lets see. ( 2sin(x)-1)(2sin(x)+1)

    then i can cancel out 2sin(x)+1 which leaves 2sin(x)-1=2sin(x)-1

    which equals 1.

    And that is the idea of an identity ?
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  7. #7
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    Quote Originally Posted by Kromletch View Post
    Lets see. ( 2sin(x)-1)(2sin(x)+1)

    then i can cancel out 2sin(x)+1 which leaves 2sin(x)-1=2sin(x)-1

    which equals 1.

    And that is the idea of an identity ?
    \displaystyle\frac{4sin^2x-1}{2sinx+1}=2sinx-1

    The idea of the identity is that the above expression is equal for all x.
    Hence, instead of using the fraction on the left in calculations,
    the right side, which is simpler, may be used instead.

    You are asked to show they are equal.
    If they are equal, then the denominator must divide into the numerator,
    hence the numerator should have the denominator as a factor.
    The 2nd factor should be the part on the right.
    That's the clue to work it through.

    4sin^2x-1=(2sinx+1)(2sinx-1)\;\;?

    4sin^2x-1=2sinx(2sinx-1)+(1)(2sinx-1)\;\;?

    4sin^2x-1=4sin^2x-2sinx+2sinx-1\;\;?

    Yes, they are equal.
    Hence, the fraction may be replaced with

    2sinx-1
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  8. #8
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    Hmm im starting to understand, ive ran into yet again another different situation.

    So (sinx-cosx)(sinx+cosx)=1-2cosx^2
    Multiply them
    sinx^2+sinxcosx-sinxcosx-cosx^2=1-2cosx^2
    for
    sinx^2-cosx^2=1-2cosx^2

    and am again unsure of where to go from here, im looking in my book at pyhagrean identities seems similar
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  9. #9
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    Quote Originally Posted by Kromletch View Post
    Hmm im starting to understand, ive ran into yet again another different situation.

    So (sinx-cosx)(sinx+cosx)=1-2cosx^2
    Multiply them
    sinx^2+sinxcosx-sinxcosx-cosx^2=1-2cosx^2
    for
    sinx^2-cosx^2=1-2cosx^2

    and am again unsure of where to go from here, im looking in my book at pyhagrean identities seems similar
    You made good progress with that.

    sin^2x-cos^2x=1-2cos^2x\;\;?

    There are a few ways to continue..

    You have "difference of squares" on the left, but you have already expanded that out.

    Notice that we have 1 on the right, so we could use the identity

    sin^2x+cos^2x=1, which is a Pythagorean Identity, as you suspected...

    This gives

    sin^2x-cos^2x=sin^2x+cos^2x-2cos^2x\;\;?

    cos^2x-2cos^2x=(1-2)cos^2x=-cos^2x

    sin^2x-cos^2x=sin^2x-cos^2x\;\;?

    True
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