# Unit Circle Proofs

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.

Prove that AE = Tan($$\theta$$)

Similar triangles may help.

2. Homework Equations
Basic similar triangles equations of:

$$\frac{OF}{AC}$$ = $$\frac{OE}{AC}$$ = $$\frac{OA}{FE}$$

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($$\theta$$) and EC = OE - cos($$\theta$$)

AE = $$\sqrt{(AC)^{2} + (EC)^{2}}$$
I need to rearrange it somehow so I get: AE = $$\sqrt{\frac{sin^{2}(\theta)}{cos^{2}(\theta)}}$$ 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.

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Hmm. I got an email saying that Integral had replied but nothing is showing here.

tiny-tim
Homework Helper
Welcome to PF!

Hi Einstein2nd! Welcome to PF! 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. This is actually the reason why it's called the tangent … it's the length of the tangent!
(… the clue's in the name … )

Alternatively, using similar triangles as suggested:

Hint: OAE is similar to OCA. Last edited:
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!

tiny-tim
Homework Helper
oops!

Hi Einstein2nd! 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. Sorry! "opposite/adjacent" for tan is right! (I've edited my previous post to correct this.)

How are you doing with the similar triangles?

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.