$[4$ pts$]$ The following problems are relevant to a mechanism that translates one kind of

motion, extension of a hydraulic cylinder, into movement of another point, such as the

hand of a robot. To design that portion of the robot, you would need to understand how the two motions are related. These calculations involve trigonometry of triangles. The basic relations, which you have no doubt seen before, are as follows:

Right Triangle

$c^2=a^2+b^2$

$cos=\frac{a}{c}$

$sin=\frac{b}{c}$

$tan=\frac{b}{a}$

General Triangle

Law of sines

$sin\frac{}{a}=sin\frac{}{b}=sin\frac{}{c}$

Law of cosines

$c^2=a^2+b^2-2$abcos$$

The configuration shown is for the problems below. Solid part BCD can pivot about point $C.$ The part $AB$ can extend $($it represents a hydraulic cylinder$).$ One end is fixed at point A$,$ and the other end is attached to point B$.$

$($a$)$ Point D moves down by $50$ mm $($and also to the left somewhat$).$ Determine the angle by which the leg CD of BCD has rotated about point C$.($Answer: $7\mathrm{.}18)$

$($b$)$ BC also rotates by $7\mathrm{.}18$ clockwise about C $($because BCD is a solid body$).$ How does the position of point B move horizontal and vertically? $($Answers: B moves $75$ mm to the right, and $4\mathrm{.}70$ mm downward.$)$

$($c$)$ Given that the point B moves $75$ mm to the right and $4\mathrm{.}70$ mm downward, by how

much has AB changed length? By how much has AB rotated $($Answers: $64\mathrm{.}7mm,5\mathrm{.}75.)$