<s>[INST] What is the angle between the median and the bisector? In a triangle ABC, what is the angle $\theta$ between the median AM and the bisector AD?
I want a way to know the measure of that angle, given the lenghts of the sides and angles of the triangle. I tried solving the triangle, with the sine and cosine laws, and I arrived at
$$
\theta = \frac{A}{2}
       - \arcsin\left(\frac{2\sqrt{(a/2)^2+b^2
                                               -ab\cos C}}
                                  {a\sin C}\right).
$$
I'm not even totally sure if it's correct and it's a bit unwieldy... I am wondering if there is a smarter aproach that gives a nicer formula. Thank you!
 [/INST] By law  of sines:
$$
\frac{\sin CAM}{\sin BAM}=\frac{\sin C}{\sin B}=:\lambda.
$$
Setting $\sin BAM=x$, and solving the resulting equation for cosine of the sum:
$$
\sqrt{1-x^2}\cdot\sqrt{1-\lambda^2x^2}-x\cdot\lambda x=\cos A\\
$$
one obtains:
$$
\sin BAM=\frac{\sin A}{\sqrt{1+\lambda^2+2\lambda\cos A}}
=\frac{\sin A\sin B}{\sqrt{\sin^2 B+\sin^2 C+2\sin B\sin C\cos A}}
$$
and similarly 
$$
\sin CAM=\frac{\sin A\sin C}{\sqrt{\sin^2 B+\sin^2 C+2\sin B\sin C\cos A}}.
$$
Correspondingly:
$$
\cos BAM=\frac{\sin B\cos A+\sin C}{\sqrt{\sin^2 B+\sin^2 C+2\sin B\sin C\cos A}},\\
\cos CAM=\frac{\sin B+\sin C\cos A}{\sqrt{\sin^2 B+\sin^2 C+2\sin B\sin C\cos A}}.
$$
Writing:
$$\small{
\cos(BAM-CAM)=\frac{(\sin B\cos A+\sin C)(\sin B+\sin C\cos A)+\sin B\sin C\sin^2A}{\sin^2 B+\sin^2 C+2\sin B\sin C\cos A}\\
=\frac{(\sin^2 B+\sin^2 C)\cos A+2\sin B\sin C}{\sin^2 B+\sin^2 C+2\sin B\sin C\cos A}
}
$$
and noticing that $|\widehat{BAM}-\widehat{CAM}|$ is the doubled value of the angle in question one finally obtains:
$$
\theta=\frac{1}{2}\arccos\frac{(\sin^2 B+\sin^2 C)\cos A+2\sin B\sin C}{\sin^2 B+\sin^2 C+2\sin B\sin C\cos A}.
$$
</s>