Very cool!! I'm actually a biologist and statistician by trade (yeah, weird combination) and have had all sorts of algebra, calculus, and advanced calculus classes (and forgot much of what I'd learned), but I'll agree that the most fun class I had was my 10th grade geometry class. BUT I don't recall learning about the Law of Sines. It would be fun to find an advanced geometric class at my local junior college.

Just to clarify, the Law of Sines is

d = a / SIN(A) = b / SIN(B) = c / SIN(C)
where a, b, c are side lengths and A, B, C are the angles opposite of the smaller case sides. On top of that, d is the diameter of the circle that can be drawn around the triangle touching all three corners (real cool!!).

OK, lets see if I can walk us through a scenario:

There are three points of interest:
1-My location at potition #1
2-My location at position #2
3-Landmark

1. While at #1 I take a bearing on the landmark and get 332 degrees (kind of NNW).

2. I then walk on a bearing of 33 degrees (ENE) for 1.5 miles and then take another bearing on the landmark, and get 301 degrees.

The first triangle angle is (360-332)+33=61 degrees.
(360-332)=28 gives the angle west from north, and then 33 gives the angle on the east side of north.

The second triangle corner is 180-(360-301)-33 = 88 degrees.
(360-301) = 59 gives the angle west of north, 33 is the angle east of north, and these two angles plus the angle for the second triangle corner have to add up to 180 degrees. That is the half arc of a circle along the line of my 33 degree bearing line of travel: 88+59+33 = 180.

So far I know two angles for the triangle: 61 degrees and 88 degrees. The third angle must be 180-61-88 = 31 degrees.

So using the Law of Sines we have

1.5 / SIN(31) = b / SIN(88) = c / SIN(61)

where b is the distance from the landmark to postion #1 and c is the distance from the landmark to position #2.

Using some algebra:
b = 1.5 x SIN(88) / SIN(31) = 2.91 miles from landmark to #1

c = 1.5 x SIN(61) / SIN(31) = 2.55 miles from landmark to #2

Lessons learned - besides the Law of Sines itself:
1 - geometry skills are useful for geometry
2 - bring a decent pad of paper and a good pen/pencil
3 - bring a small solar-powered calculator that does trig functions in degrees (as opposed to radians)
4 - having a good 6" unbreakable ruler would help too

Thanks Benjammin!

P.S. My wife has been watching me with pieces of paper, a ruler, and a protractor on my lap drawing pictures & triangles and thinks (knows?) I am nuts. Nothing new. She has accepted my odd hobbies.

{I'd like to self-nominate this post for the "most math in an ETS post" award} <img src="/images/graemlins/blush.gif" alt="" />