We sketch a right triangle. The height of the short side of the triangle is 500 feet. That is the difference between elevations from our tent to the top of Coyote Hill. The length of the long side of the right triangle is 2640 feet, the distance on our map from tent to hilltop. We connect the free end of the long side to the free end of the short side of the triangle.

The angle forming this hypotenuse is the altitude of the sun when it first shines on our tent.  To find that angle we divide 500 by 2640 to get 0.1894, the tangent of the angle we seek. In trigonometry tables or with a trigonometry calculator, we find the angle with a tangent of 0.1894. That is, we find the arctangent (ATAN) of 0.1894. So, the angle is 10.7 degrees.

When will the sun reach an altitude of 10.7 degrees? And will it be shining from due east or will it be slightly north or south of east? To obtain data for the sun, we go to the Naval Observatory Web site. Our map shows our fictional Cold Gulch to be very near the town of Hill City, so we use that location and the date 2006 August 15. We find the sun will be at altitude 10.7 degrees at 6:04 AM MST. But the sun will not be due east at that time. It will be 9.3 degrees north of east.

From our topographic map we find the elevation of Coyote Hill at the new direction, measure the distance to it on the map and recalculate the angle. And we find the sun will warm our tent at about the same time. However, if we hike a mile further up Rock River before setting up camp, we find that Coyote Hill is higher and closer to our campsite. So there we would have to wait till 7:15 AM for the sun to warm our tent.

We don't have to spend a lot of time doing these calculations. When we become familiar with the apparent movements of the sun and with topographic maps, we can gain a close idea of the possibilities for campsites without detailed data. We can check two or three sets of figures and make our decisions. Other factors will be involved in choosing a campsite. The time we spend with the trigonometry will be only a few minutes. On our hiking trip, our time will be returned to us.

The trigonometry amounts to little more than dividing one number by another and finding the result in a trigonometric table or calculator, with a pocket calculator, or with a computer spreadsheet. With that done, on a night hike or weeklong backpacking trip we can obtain some useful details about when and where moonlight and sunlight will ease our way

When planning a camping trip into the wilderness, a little trigonometry can give us a good idea of where we may locate our tent so that it is in the shade in the hot evenings but in the sunshine in the chilly early mornings. Or we can place it so that it is surrounded by moonlight at night. For a night hike we can determine when sections of the trail we hike will be lit by the moon and at what time of night.

If our topographic map indicates the trail passes through a potentially dark ravine, we can determine when the moon will illuminate that area, thus making our hike through it less precarious.

The time we spend doing a little trigonometry will be saved later when we find that any dew collecting upon our tent overnight will be dried early in the morning. Or if we are camping in higher mountains such as the Bighorns, the afternoon shade will deactivate the mosquitoes and allow our tent to cool enough to be habitable.

While other hikers are burning precious body energy shivering as they pack their camping gear or wait for their tent to dry before the day’s journey, we can have dry tents rolled up and strapped to our backpack as we relax in the sunshine to eat breakfast. We spent less time doing our trigonometry than they do shivering and waiting on the sunlight. We are ready to go.

For night hikes this same principle can be used to determine when a trail may be illuminated by the moonlight. If we are hiking west across Black Elk Wilderness at night and using Trail #7 on its ascent of Harney Peak, there is a short chasm which that trail goes through before ascending the peak. We can use the azimuth and altitude of the moon, apply trigonometry to the surrounding hills and gain a good estimate of when we want to hike through that chasm.

Some may say, sure that sounds good on paper, but does it really work? Or why waste the time? Well, of course it doesn’t work on cloudy nights or overcast mornings. Fortunately nature is not so predictable as to remove all elements of adventure from our journey.

But then neither are we so careless as to go stumbling along in the dark thinking we can bluff the wilderness into letting us through. And we are confident our geometry and physics teachers did not consider it a waste of time when they taught us this discipline. So we nurture their investment like Jefferson would, by applying our science to nature.

But of course anyone hiking at night does well to have a flashlight handy in case nature places a red checkmark by his or her trigonometry problem. As a matter of fact, these methods are effective and good practice in trigonometry, which actually has many applications in daily life. And it is a gratifying experience to round a corner in the trail at night and find the full moon flooding that precarious chasm with light.