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Unit Circle Proofs

  • #1
1. Homework Statement
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.

http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg

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. Homework 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:

Answers and Replies

  • #2
Hmm. I got an email saying that Integral had replied but nothing is showing here.
 
  • #3
tiny-tim
Science Advisor
Homework Helper
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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:
 
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  • #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!
 
  • #5
tiny-tim
Science Advisor
Homework Helper
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249
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?
 
  • #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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