Let $AB$ is rotating at $\omega_{AB}=4$ rad/s. Find $\omega_{CD}$ when $\theta=\pi/6$.
So the first thing I did was wrote an express for $CD$ call it $r$. $\phi$ is Angle $CAB$ for reference.
By law of sines I have that $r=\dfrac{0.3\sin \phi}{\sin \theta}$
Since $\Delta ABC $ is a right triangle, $r=0.6\cos \theta$ Taking derivative, and equating to each other I find that $\omega_{CD}$ is $\dfrac{0.3\cos \phi\cdot \omega_{AB}}{r\cos \theta - 0.6(\sin \theta)^2}$
For some reason this isn't the answer. Did I use the wrong relationship?
According to the sine law you get:
$$ \frac { AB }{ \sin { \theta } } \quad =\quad \frac { AC }{ \sin { (\pi -(\theta +\phi )) } } \\ \frac { AB }{ \sin { \theta } } \quad =\quad \frac { AC }{ \sin { \left( \theta +\phi \right) } } \\ \sin { \theta } \quad =\quad \frac { AB }{ AC } \sin { \left( \theta +\phi \right) } \\ \frac { d }{ dt } \sin { \theta } \quad =\quad \frac { AB }{ AC } \frac { d }{ dt } \sin { \left( \theta +\phi \right) } \\ \frac { d\theta }{ dt } \cos { \theta } \quad =\quad \frac { AB }{ AC } \left( \frac { d\theta }{ dt } +\frac { d\phi }{ dt } \right) \cos { \left( \theta +\phi \right) } \\ { \omega }_{ CD }\cos { \theta } \quad =\quad \frac { AB }{ AC } \left( { \omega }_{ CD }+\omega \right) \sqrt { 1-\sin ^{ 2 }{ \left( \theta +\phi \right) } } \\ { \omega }_{ CD }\cos { \theta } \quad -\frac { AB }{ AC } { \omega }_{ CD }\sqrt { 1-\sin ^{ 2 }{ \left( \theta +\phi \right) } } \quad =\quad \frac { AB }{ AC } \omega \sqrt { 1-\sin ^{ 2 }{ (\theta +\phi ) } } \\ { \omega }_{ CD }\left( \cos { \theta } -\frac { AB }{ AC } \sqrt { 1-{ \left( \frac { AC }{ AB } \sin { \theta } \right) }^{ 2 } } \right) \quad =\quad \frac { AB }{ AC } \omega \sqrt { 1-{ \left( \frac { AC }{ AB } \sin { \theta } \right) }^{ 2 } } \\ { \omega }_{ CD }\quad =\quad \frac { \frac { AB }{ AC } \omega \sqrt { 1-{ \left( \frac { AC }{ AB } \sin { \theta } \right) }^{ 2 } } }{ \cos { \theta } -\frac { AB }{ AC } \sqrt { 1-{ \left( \frac { AC }{ AB } \sin { \theta } \right) }^{ 2 } } } $$
Now plug in the constants and the value of $\theta$ and you'll get the answer