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Math Help - Deriving identities

  1. #1
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    Deriving identities

    Okay, so I have a possibly sort of unique situation!

    My instructor does not give any identities or formulas for the tests, and while my final is a long ways off I need to do this at least once now that I'm done with this chapter.

    This is a whopper, but I know that once I can see the relationships between them I'll have an easy time with it.

    The sets of identities I need to be able to derive from scratch:

    -Cosine, Sine, and Tangent of a sum or difference (6 identities)
    -Double angle identities of a cosine (3 identities), sine (1) and tangent (1)
    -( I have the cofunction identities, they're easy enough)
    -Product to sum identities (4)
    -Sum to product identities (4)
    and lastly the half angle identities (5)

    In total that's 30 identities, and I have 6 memorized so 24 identities I have to be able to derive to be able to use on my final exam. This is an online class ( tests are proctored ) and I am allowed to use my TI-89. ( That 'solve(' function is such a time saver )

    I have all of the identities written down in my notes ( which are in notepad, so they're ugly. I have dysgraphia and taking written notes is an exercise in futility for me ) I've attached the text document with all my identities for this section

    Where do I start? Which ones do I memorize and which ones do I derive?
    Attached Files Attached Files
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  2. #2
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    Quote Originally Posted by Wolvenmoon View Post
    Okay, so I have a possibly sort of unique situation!

    My instructor does not give any identities or formulas for the tests, and while my final is a long ways off I need to do this at least once now that I'm done with this chapter.

    This is a whopper, but I know that once I can see the relationships between them I'll have an easy time with it.

    The sets of identities I need to be able to derive from scratch:

    -Cosine, Sine, and Tangent of a sum or difference (6 identities)
    -Double angle identities of a cosine (3 identities), sine (1) and tangent (1)
    -( I have the cofunction identities, they're easy enough)
    -Product to sum identities (4)
    -Sum to product identities (4)
    and lastly the half angle identities (5)

    In total that's 30 identities, and I have 6 memorized so 24 identities I have to be able to derive to be able to use on my final exam. This is an online class ( tests are proctored ) and I am allowed to use my TI-89. ( That 'solve(' function is such a time saver )

    I have all of the identities written down in my notes ( which are in notepad, so they're ugly. I have dysgraphia and taking written notes is an exercise in futility for me ) I've attached the text document with all my identities for this section

    Where do I start? Which ones do I memorize and which ones do I derive?
    Everything can be derived from cos(A + B) = ..... and sin(A + B) = .....

    However, derivations take time and may themselves require memorisation ....
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  3. #3
    Senior Member pacman's Avatar
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    this may help, the WEB have plenty of it only IF you try browsing,
    this one is the best: Trigonometric Identities -- from Wolfram MathWorld
    also this List of trigonometric identities - Wikipedia, the free encyclopedia

    Double-, triple-, and half-angle formulae
    See also: Tangent half-angle formula
    These can be shown by using either the sum and difference identities or the multiple-angle formulae.
    Double-angle formulae[16][17]Triple-angle formulae[18][14]Half-angle formulae[19][20]

    The tangent half-angle formulae are as follows. Let
    Then we have
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  4. #4
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    Okay, so I'm working on memorizing these by working the identities. Right now I'm on the double angle identities. What I derived are:

    (Latex never ceases to humiliate me, so they're in calculator terms)

    cos2A = cos^2(A) - sin^2(A)

    sin2A = 2sin(A)*cos(A)

    tan2A = (2tan(A))/(1-(tan^2(A))

    There are two others for cosine,

    Cos2A = 2*cos^2(A) - 1
    cos2A = 1 - 2sin^2(A)

    This would mean that:
    cos^2(A) - sin^2(A) = 2*cos^2(A) - 1

    Where I'm stuck is -sin^2(A) becomes cos^2(A) - 1, and how cos^2(A) becomes -sin^2(A)

    This is something to do with the Pythagorean identity, but I'm not seeing it.
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  5. #5
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    Quote Originally Posted by Wolvenmoon View Post
    Okay, so I'm working on memorizing these by working the identities. Right now I'm on the double angle identities. What I derived are:

    (Latex never ceases to humiliate me, so they're in calculator terms)

    cos2A = cos^2(A) - sin^2(A)

    sin2A = 2sin(A)*cos(A)

    tan2A = (2tan(A))/(1-(tan^2(A))

    There are two others for cosine,

    Cos2A = 2*cos^2(A) - 1
    cos2A = 1 - 2sin^2(A)

    This would mean that:
    cos^2(A) - sin^2(A) = 2*cos^2(A) - 1

    Where I'm stuck is -sin^2(A) becomes cos^2(A) - 1, and how cos^2(A) becomes -sin^2(A)

    This is something to do with the Pythagorean identity, but I'm not seeing it.
    Remember that sin^2(\theta)+cos^2(\theta)=1

    So you can solve this for sin^2(\theta)=1-cos^2(\theta) and substitute. The negative of this is just cos^2(\theta)-1
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  6. #6
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    I accept with information: cos2A = cos^2(A) - sin^2(A)

    sin2A = 2sin(A)*cos(A)

    tan2A = (2tan(A))/(1-(tan^2(A))

    There are two others for cosine,

    Cos2A = 2*cos^2(A) - 1
    cos2A = 1 - 2sin^2(A)

    This would mean that:
    cos^2(A) - sin^2(A) = 2*cos^2(A) - 1
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