Trigonometric Analysis: Calculating Segment Lengths in a Broken Tree Problem

In the diagram below, a point \(P\) is on horizontal ground, \(10\) m from a vertical tree, \(TS\).

The angle of depression of \(P\) from the top of the tree is \(50{}^\circ \).

Due to a strong wind, the tree was broken and the angle of depression of \(P\) from \(R\), the top of the remaining part of the tree, became \(30{}^\circ \).

Find the length of the broken part of the tree, \(TR\).

Correct your answer to \(3\) significant figures.