Unit Circle Proofs

1. Apr 1, 2008

Einstein2nd

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.

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

Similar triangles may help.

2. Relevant 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.

2. Apr 1, 2008

Einstein2nd

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

3. Apr 1, 2008

tiny-tim

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: Apr 2, 2008
4. Apr 1, 2008

Einstein2nd

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. Apr 2, 2008

tiny-tim

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!

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

How are you doing with the similar triangles?

6. Apr 2, 2008

Einstein2nd

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.