1. ## computing angle and radius with few infos

hello!
i have a little question about a trigonometry exercice. and i'm stucked getting the formulas
is it possible to get the angle and the radius of a circle if i have only the arc length and the chord length of these

2. Originally Posted by mantus
hello!
i have a little question about a trigonometry exercice. and i'm stucked getting the formulas
is it possible to get the angle and the radius of a circle if i have only the arc length and the chord length of these
hello , i don think so .

Let A and B be the arc length and chord length respectively , where their values are known .

$\displaystyle A=r\theta$ ---1

$\displaystyle B=2r\sin \frac{\theta}{2}$ ---2

If you divide the simultaneous equations to solve for $\displaystyle \theta$

$\displaystyle \frac{A}{B}=\frac{r\theta}{2r\sin \frac{\theta}{2}}$

$\displaystyle \frac{A}{B}=\frac{\theta}{2\sin \frac{\theta}{2}}$

For convenience , let $\displaystyle \frac{A}{B}=1$

so $\displaystyle \theta=2\sin (\frac{\theta}{2})$ which i don see how to solve this .

3. Originally Posted by mantus
hello!
i have a little question about a trigonometry exercice. and i'm stucked getting the formulas
is it possible to get the angle and the radius of a circle if i have only the arc length and the chord length of these
Yes.

The circumference of a circle is $\displaystyle 2\pi r$.

So the arc length $\displaystyle a$ is $\displaystyle \frac{2\pi r \theta}{360} = \frac{\pi r \theta}{180}$ (if the angle is measured in degrees)

or

$\displaystyle \frac{2\pi r \theta}{2\pi} = r\theta$ (if the angle is measured in radians).

Let's assume that the angle is measured in radians.

You can also create an isosceles triangle using the chord and 2 radii.

The angle between them is $\displaystyle \theta$.

If we call the chord length $\displaystyle l$, then by the Cosine Rule we have

$\displaystyle l^2 = r^2 + r^2 + 2r\cdot r \cos{\theta}$

$\displaystyle l^2 = 2r^2 + 2r^2\cos{\theta}$.

Now you have two equations in two unknowns.

Edit: Upon further inspection it doesn't look like these two equations can be solved simultaneously for $\displaystyle r$ and $\displaystyle \theta$.