# Thread: Four segments, find angle?

1. ## Four segments, find angle?

Given the geometry shown below I need to find angle r. I now all the sides (a,b,c,d). I do NOT know what angle K is. Here is the geometry setup:

I approached this problem the following way:
EC = AB + 2*BC*sin(k)
DE^2 = DC^2 + EC^2 - 2*DC*EC*cos(r)
I have two equations and three unknowns. Which means I am missing one more equation. Any suggestions? Or maybe a better approach?
Thanks.

2. Let the H be the foot of the perpendicular from D to the line AB produced on the side of A, and let I be the foot of the perpendicular from C to the line AB produced on the side of B. Then the angles HDA and ICB are equal to κ, and HI = $b\sin{\kappa}+d+a\sin{\kappa}=c\cos{r}$.

$\therefore\ \sin{\kappa}\ =\ \frac{c\cos{r}-d}{b+a}\quad\ldots\fbox{1}$

Also if DH meets the line EC produced on the side of E at J, then DJ =
$(b-a)\cos{\kappa}=c\sin{r}$.

$\therefore\ \cos{\kappa}\ =\ \frac{c\sin{r}}{b-a}\quad\ldots\fbox{2}$

From [1] and [2], κ can be eliminated.

3. Thanks a lot JaneBennet. I see what I was missing there
Originally Posted by JaneBennet
From [1] and [2], κ can be eliminated.
I used this formula to connect cos(k) and sin(k):
$
\cos{\kappa}\ =\ \sqrt{1- \sin{\kappa}^2}$
$\quad\ldots\fbox{3}
$

After substitution I get a formula that has $\sin{r}^2$, $\cos{r}^2$ and $sin{r}$. This makes it tricky to get actual r. Any suggestions?

4. After playing with the equations I boiled it down to this formula:
$2abc^2(\sin{r}^2-\cos{r}^2)-2d\cos{r}(b-a)^2 = -(b^2c^2+a^2c^2+d^2(b-a)^2) \quad\ldots\fbox{4}$ .
Everything on the right side is a constant. But how do I convert the difference of squares there? It would be convenient if am able to collapse it to $\cos{r}^2$, but how?

5. It’s very complicated, I know. But if you do everything carefully, you should end up with a quadratic equation in $\cos{r}$, which can always be solved using the quadratic formula.

6. ## More calculations

It's more interesting than I thought I tried another approach for substitution, which lead me pretty much to the same result, meaning I still have $\sin{r}^2-\cos{r}^2$. Here are my calculations. First I simplify both sides of the equations and then I bring them together. I checked the signs a few times and they all match properly. Any suggestions?

Thanks a lot.

7. I did find the solution with the help of Mathcad. Thanks a lot for the help.