Atlas 750 makes a very cogent point...you must pay attention to the signs of latitude and declination. As he points out, Southerly declinations are negative, as are longitudes East of the Prime Meridian (Greenwich), and southerly latitudes as well. The cosine formula is known by heart by all marine navigators worthy of the name.
The conscientious ones use it several times daily. I recall during my time as a merchant mariner, the Mate would take a bearing on the setting sun from the gyro-repeater on the bridge-wing, then compute from the almanac what the bearing should be given time and DR latitude of the observation.
The difference between the two gave a gyrocompass-error, and was a standard practice. Compass-errors are taken every watch, usually as soon as the watchkeeper assumes control, and about halfway through his watch. The sun or a star (or planet) can be used so observations of compass-error can be made at all times when celestial objects are visible. Best results are obtained with objects relatively low in the sky.
The compass-errors were recorded in a book kept readily to hand on the chart-table, and the Masters Night Orders usually contained the phrase "compass errors to be obtained."
"Simple sums for simple sailors" is the essence of celestial navigation. In this age of instant positions via GPS it is heartening to see someone looking to the heavens and thinking "why is this so"...and having sufficient interest to ask the question.