Geometry question with the perimeter of a circle

[Jathor]

say I have a circle with a diameter of d and there is a point at the top of the circle, p. I want to know the distance from point p to any other point on the perimeter and the angle θ from the tangent line of p. is there a function that will describe this?

I will try to put up a picture to better illustrate.

 

[Jathor]

https://C:\Users\Candice\Pictures\photo.JPG

 

[eumyang]

[ IMG ]https://C:\Users\Candice\Pictures\photo.JPG[ /IMG ]

You can't put up images this way. If the picture is small enough, under the editing window, under "Additional Options," click "Manage Attachments."

 

[Vorde]

Yay I get to use some of my obscure math knowledge!

In the early days of trigonometry there were lots of different trig functions (Instead of the 3 main and 3 inverse functions we have now). Amongst these now-obsolete functions was a function $crd(\theta)$ (meaning chord).

$crd(\theta)$ was defined as $$2sin(\frac{\theta}{2})$$
If r is the radius of your circle, the distance from the point at the top to any other point on the circle is $rcrd(\theta)$ where $\theta$ is the angle between the point at the top and the point you want to know the distance to as measured from the center of the circle. This angle can be related to the angle from the tangent line by simple geometry.

I hope this helps!

 

[Jathor]

thanks Vorde, that helped a lot (: sorry i couldn't get the picture working but you got it.

 

[Mute]

Yay I get to use some of my obscure math knowledge!

In the early days of trigonometry there were lots of different trig functions (Instead of the 3 main and 3 inverse functions we have now). Amongst these now-obsolete functions was a function $crd(\theta)$ (meaning chord).

$crd(\theta)$ was defined as $$2sin(\frac{\theta}{2})$$
If r is the radius of your circle, the distance from the point at the top to any other point on the circle is $rcrd(\theta)$ where $\theta$ is the angle between the point at the top and the point you want to know the distance to as measured from the center of the circle. This angle can be related to the angle from the tangent line by simple geometry.

I hope this helps!

Some latex notes: it helps to distinguish functions from variables by making the names of functions non-italicized. For sine there is a built in function: \sin (there are similar ones for tangent, cosine, ln, etc). For non-standard functions, such as crd, you can use the \mbox command, e.g., \mbox{crd}.

This will render it as

$$\mbox{crd}\theta.$$

I point this out because when you wrote "$rcrd(\theta)$", I was momentarily confused, thinking that you were using a modified chord function there. $r\mbox{crd}\theta$ is immediately clear. ;) (Also, I am not using parentheses because the non-italicized names of functions makes it clear what's the function and what's the argument, at least in this case of just one argument).

 

[Vorde]

That's awesome help!

I'm trying to teach myself as much latex as possible for ease of writing (I've been typing $[/ite... for 5 months until last week I figured out the #.# thing), thanks for the tip!$

 

[eumyang]

$$\mbox{crd}\theta.$$

Even better would be to insert spaces. A slash followed by a comma, colon, or semicolon will give you a space, depending on the width. So instead of
$rcrd(\theta)$.
one could write
$r\,\mbox{crd}\,\theta$.