This page was added to this site to illustrate an answer posted on AllExperts.com, and is not meant to be a thorough discussion of the PZS triangle. It will be updated and expanded at some future date, with the discussion below as merely one example of how to use the triangle.
The question posed was, what angle does the Sun rise at relative to the horizon when it has a particular declination? The diagram shown below describes the so-called PZS triangle required to answer that question:
Here is the PZS triangle formed by the rising Sun:
(1) the arc from the Zenith to the Celestial Pole, which is equal to 90 - the observer's latitude
(2) the arc from the Celestial Pole to the Sun, which is equal to 90 - the Sun's declination
(3) the arc from the Zenith to the rising Sun, which is 90 degrees.
In this case, S is the angle made at the Sun by the vertical circle from the Sun to the zenith, and the hour circle from the Sun to the Celestial Pole. If we call that angle S, then the angle the Sun rises at relative to the horizon is also S, as shown in the diagram below. (The Sun's declination circle is also the path it follows while rising, and is at a right angle to the hour circle through the Sun.)
Above, a PZS triangle (usually meaning Pole-Zenith-Star, but in this case Pole-Zenith-Sun, since it is showing the Sun rising on the eastern horizon). Straight lines are used in place of curved arcs, but the description of the angles is correct. The line from the Pole to the Sun represents the hour circle corresponding to the current right ascension of the Sun, and the Sun is rising at right angles to that hour circle, moving upwards along the declination circle corresponding to its current declination. The angle from the Pole to the Sun to the Sun's rising path is a right angle, and the angle from the Zenith to the Sun to the Horizon is also a right angle, so the angle the Sun rises at relative to the horizon (shown as angle S) is the same as the angle at the Sun between its hour circle and the vertical circle from the Sun to the zenith. In the PZS triangle, all three sides are known (as shown by their labels), and as a result it is a relatively simple matter of spherical trigonometry to solve for the three angles.