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Unit Circle Proofs

  1. Apr 1, 2008 #1
    1. The problem statement, all variables and given/known data
    The problem comes with a diagram but I'll use the wikipedia diagram because it's nice and pretty and I'll just rearrange the letters to suit it.

    [​IMG]

    Just in case the image doesn't load in the page: http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg

    Prove that AE = Tan([tex]\theta[/tex])

    Similar triangles may help.

    2. Relevant equations
    Basic similar triangles equations of:

    [tex]\frac{OF}{AC}[/tex] = [tex]\frac{OE}{AC}[/tex] = [tex]\frac{OA}{FE}[/tex]


    3. The attempt at a solution

    After working out the above relationship, I've tried to look specifically at the ACE triangle. I already have AC = sin([tex]\theta[/tex]) and EC = OE - cos([tex]\theta[/tex])

    AE = [tex]\sqrt{(AC)^{2} + (EC)^{2}}[/tex]
    I need to rearrange it somehow so I get: AE = [tex]\sqrt{\frac{sin^{2}(\theta)}{cos^{2}(\theta)}}[/tex] but I'm not sure how to go about this. I'm getting a bit lost/sidetracked so hopefully you can give me some help. There a part (b) in this question but I won't ask about that until after this and hopefully once I can solve this (b) will be easy.

    Thanks in advance for your help.
     
  2. jcsd
  3. Apr 1, 2008 #2
    Hmm. I got an email saying that Integral had replied but nothing is showing here.
     
  4. Apr 1, 2008 #3

    tiny-tim

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    Welcome to PF!

    Hi Einstein2nd! Welcome to PF! :smile:

    One definition of tan is: "opposite/adjacent".

    In this case, in the triangle OAE, the side opposite theta is AE, and the side adjacent to theta is OA.

    So tantheta = AE/OA = AE/1 = AE. :smile:

    This is actually the reason why it's called the tangent … it's the length of the tangent!
    (… the clue's in the name … :wink:)

    Alternatively, using similar triangles as suggested:

    Hint: OAE is similar to OCA. :smile:
     
    Last edited: Apr 2, 2008
  5. Apr 1, 2008 #4
    I may be on the wrong track here but you are saying OE = tan(theta) when I was trying to prove AE = tan(theta). Has there been a mixup with the letters on the wikipedia diagram I gave or is there somethign more to it than that?

    Thank you for your help so far!
     
  6. Apr 2, 2008 #5

    tiny-tim

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    oops!

    Hi Einstein2nd! :smile:

    Yes, you're right … I got the letters wrong.

    I should have written:

    In this case, in the triangle OAE, the side opposite theta is AE, and the side adjacent to theta is OA.

    So tantheta = AE/OA = AE/1 = AE. :smile:

    Sorry! :redface:

    "opposite/adjacent" for tan is right! :smile:

    (I've edited my previous post to correct this.)

    How are you doing with the similar triangles?
     
  7. Apr 2, 2008 #6
    All done and proven! I then went on to prove Sec, Cosec and Cot! I'll post answer later as I don't have them on me. Proving Tan wasn't that hard in the end. Similar triangles made it very easy.
     
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