# Thread: complicated geometry in mechanics

1. ## complicated geometry in mechanics

I have to determine the distance along the Z-axis from O to where the upper line crosses the Z-axis and makes angle $\beta$ between the Z-axis and the line.
The lower line which length equals L starts at the origin and makes an angle $\alpha$ between the Z-axis and the line itself (so the angle between the line and the X-axis is $\frac{1}{2}\pi-\alpha$)

This distance along the Z-axis equals (Lcos $\alpha$ + Lsin $\alpha$cotan $\beta$)ez
ez is the unit vector and there should be an arrow above the letter e.
I don't understand why you can express this distance like this.

Because I'm not certain if the rest of you can see this picture, since it's posted on a secured website (login), I have posted the same picture as an attachment.

2. Hi

See drawing attached

$z = z_0 + z_1$

$z_0 = L\: cos \alpha$

$tan \beta = \frac{L \:sin \alpha}{z_1}$

Therefore $z = L \:cos \alpha + \frac{L \:sin \alpha}{tan \beta}$

ImageShack - Image Hosting :: drawingue6.jpg

3. Great, I understand it.

Originally Posted by running-gag
Hi

See drawing attached

$z = z_0 + z_1$

$z_0 = L\: cos \alpha$
This part I understood.

$tan \beta = \frac{L \:sin \alpha}{z_1}$

Therefore $z = L \:cos \alpha + \frac{L \:sin \alpha}{tan \beta}$
This was my problem because I was confused about that cotan, I tried to solve it by using the geometric circle.

4. OK

Cotangent is only the inverse of tangent

$cotan \theta = \frac{cos \theta}{sin \theta} = \frac {adjacent \:side}{opposite \:side}$

List of trigonometric identities - Wikipedia, the free encyclopedia

5. That wasn't the problem.
I'll attach another file so you can see which figure I used, then you might understand my confusion.
It's a words-file but it just contains 1 picture (copied from a PDF-file).
I assume you can open this file, otherwise just let me know and I'll try to use Imageschack.
As you can see an arbitrary line which makes an arbitrary angle crosses Y=1 (cartesian graph with X-axis and Y-axis, unit circle with a diameter of 1) at X=cotan angle.