5. In a regular tetrahedron \( P-ABC \), \( AB=1, AP=2 \). A plane \( \alpha \) passing through \( AB \) bisects its volume. Then the cosine of the angle between edge \( PC \) and plane \( \alpha \) is \(\qquad\).
(2017, National High School Mathematics League)
Let the midpoints of $AB$ and $AC$ be $K$ and $M$, respectively. It is easy to prove that the plane $ABM$ is the plane $\alpha$.
By the median length formula, we know that $AM^2 = \frac{3}{2}$.
Thus, $KM = \sqrt{AM^2 + AK^2} = \sqrt{\frac{3}{2} - \left(\frac{1}{2}\right)^2} = \frac{\sqrt{5}}{2}$.
It is also easy to see that the projection of line $PC$ onto plane $\alpha$ is line $MK$, and $CM = 1$, $KC = \frac{\sqrt{3}}{2}$. Therefore,
$$
\cos \angle KMC = \frac{KM^2 + MC^2 - KC^2}{2 \cdot KM \cdot MC} = \frac{3 \sqrt{5}}{10}.
$$

Thus, the cosine of the angle formed by edge $PC$ and plane $\alpha$ is $\frac{3 \sqrt{5}}{10}$.
References:
[1] 2016 National High School Mathematics League Jiangxi Preliminary Contest [J]. Middle School Mathematics, 2017(5).
[2] 2016 National High School Mathematics League Zhejiang Preliminary Contest [J]. Middle School Mathematics, 2017(7).
[3] 2016 National High School Mathematics League Hubei Preliminary Contest [J]. Middle School Mathematics, 2017(5).
[4] 2016 National High School Mathematics League Gansu Preliminary Contest [J]. Middle School Mathematics, 2017(4).
[5] 2016 National High School Mathematics League Heilongjiang Preliminary Contest [J]. Middle School Mathematics, 2017(8).