# Thread: [SOLVED] Equation system to solve

1. ## [SOLVED] Equation system to solve

Hi,

Got this equation system to solve, and its just not going anywhere to me, keep banging a dead end... Hope you guys can help. The unknowns are d1 and d2, the rest is assumed as known.

Any help appreciated!

THNX!

2. Originally Posted by crapmathematician
Hi,

Got this equation system to solve, and its just not going anywhere to me, keep banging a dead end... Hope you guys can help. The unknowns are d1 and d2, the rest is assumed as known.

Any help appreciated!

THNX!
$x_1 - d_1 \sin x_1= x_2 + d_2 \cos x_2 \rightarrow(1)$

$y_1 - d_1 \cos x_1= y_2 + d_2 \sin x_2 \rightarrow(2)$

Rearrange $(1): x_1 - x_2 - d_2 \cos x_2 = d_1 \sin x_1$

$\Rightarrow d_1 = \frac{x_1 - x_2 - d_2 \cos x_2}{\sin x_1} \rightarrow(3)$

Now sub $(3)$ into $(2)$ and solve for $d_2.$

Then, plug the $d_2$ you found into $(1)$ to find $d_1.$

3. Originally Posted by Anonymous1
$x_1 - d_1 \sin x_1= x_2 + d_2 \cos x_2 \rightarrow(1)$

$y_1 - d_1 \cos x_1= y_2 + d_2 \sin x_2 \rightarrow(2)$

Rearrange $(1): x_1 - x_2 - d_2 \cos x_2 = d_1 \sin x_1$

$\Rightarrow d_1 = \frac{x_1 - x_2 - d_2 \cos x_2}{\sin x_1} \rightarrow(3)$

Now sub $(3)$ into $(2)$ and solve for $d_2.$

Then, plug the $d_2$ you found into $(1)$ to find $d_1.$

That's where the problem is! When I put (3) into (2) to solve for $d_1$ I get a beast which looks like this:

and I have no clue how to make it tidy to solve for $d_1$

4. $\frac{x_1 \cos x_1 - x_2 \cos x_1 - d_2 \cos x_1 \cos x_2}{\sin x_1} = y_2 - y_1 + d_2 \sin x_2$

$\implies x_1 \cos x_1 - x_2 \cos x_1 - d_2 \cos x_1 \cos x_2 = y_2 \sin x_1 - y_1 \sin x_1 + d_2 \sin x_2 \sin x_1$

$\implies x_1 \cos x_1 - x_2 \cos x_1 - y_2 \sin x_1 + y_1 \sin x_1 = d_2 \sin x_2 \sin x_1 + d_2 \cos x_1 \cos x_2$

$\implies x_1 \cos x_1 - x_2 \cos x_1 - y_2\sin x_1 + y_1 \sin x_1 = d_2 (\sin x_2 \sin x_1 + \cos x_1 \cos x_2 )$

$\implies d_2 = \frac{x_1 \cos x_1 - x_2 \cos x_1 - y_2 \sin x_1 + y_1 \sin x_1}{\sin x_2 \sin x_1 + \cos x_1 \cos x_2}.$

Yeah it's ugly, but I'm not sure I see the issue...?

5. Change of situation!!!

If I assume that I know d1 and d2 but do not know alpha1 and alpha2 instead. From equation 1 I have expressed sina1 which results as (x1-x2-d2*cosa2)/d1. How do I change this in order to be able to substitute for cosa1 in equation 2? Or is there another way to solve it?