Find the coordinates of points in terms of an angle

Question

Given:

1. Coordinates of point $A(0;0)$
2. Coordinates of point $C(0; y_C)$
3. Lengths $AB, CO, DO, CD$
4. Angle $\angle CDO=\frac{\pi}{2}$
5. $DL=\frac{3}{5}DO$
6. $\theta=\theta(t)$ - angle of rotation segment $AB$

Need to find coordinates of points $D, L, O, B$ in terms of $\theta$.

It's easy to find coordinates for point $B$: $$x_B=AB\cos(\theta)$$ $$y_B=AB\sin(\theta)$$

What to do with other three points?

UPD1.

It's possible to find coordinates of $M$ since we know $C$ and $B$:

$$x_M=\frac{AB\cos(\theta)}{2}$$ $$y_M=\frac{AB\sin(\theta)+y_C}{2}$$

It is also possible to find length of $CB$ and then length of $DM$: $$CB^2=(x_B-x_C)^2+(y_B-y_C)^2$$ $$CB^2=AB^2\cos^2(\theta)+(AB\sin(\theta)-y_C)^2=AB^2-2AB \cdot y_C\sin(\theta)+y^2_C$$ $$CM^2=DM^2=\frac{1}{4}CB^2$$

Find $\angle CMD$ using law of cosines:

$$CD^2=CM^2+DM^2-2CM \cdot DM\cos(\angle CMD)$$ $$CD^2=2DM^2-2DM^2\cos(\angle CMD)$$ $$CD^2=(AB^2-2AB \cdot y_C \sin(\theta)+y^2_C)\frac{1-\cos(\angle CMD)}{2}$$ $$\cos(\angle CMD)=1-\frac{2CD^2}{AB^2-2AB \cdot y_C \sin(\theta)+y^2_C}$$

But I still don't understand how to find D's coordinates in terms of $\theta$. It might be useful to use that point $D$ moves in circle of $CD$ raidus?

Answer 1

HINT.

Let $M$ be the midpoint of $CB$. Then both $DM=CM$ and $DC$ are known.