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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.


    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.
    Last edited by a moderator: Apr 23, 2017
  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


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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


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    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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