Proof of Unit Circle: AE = Tan(\theta)

[Einstein2nd]

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($$\theta$$)

Similar triangles may help.

Homework Equations

Basic similar triangles equations of:

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

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.

Thanks in advance for your help.

 

[Einstein2nd]

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

 

[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.

 

[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!

 

[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!

"opposite/adjacent" for tan is right!

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

How are you doing with the similar triangles?

 

[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.