Just wondering what level of orienteering abilities we might share out there. I've mentioned this before a while back, but does anyone know how to find themselves using only landmarks, a compass and no map?

Most orienteering is taught using a scaled map and an orienteering compass with a scale or scales calbrated to the map. This is a nice thing to have, but not absolutely necessary if you know how to triangulate your position without a but knowing a landmark. It does require you keep or memorize a sine table. The nice thing is if you take the time and effort to memorize the table, it never changes.

I've put this method to the test at least a dozen times, and I've only missed my mark on three occasions. Terrain was a factor in each miss, but I've learned to compensate for it since then. In each case, I still found my way to the destination by complementary means.
PS:

I certainly enjoy playing with map, compass, and GPS navigation skills, and I'm quite familiar with triangulation using a map and compass to identify one's position on a map (obtain bearings to three landmarks and draw direction lines on a map through the landmarks - location is where the three direction lines intersect).

Can you give more detail on the method you are talking about?? To use the sine function (or any other trig function) you'd need a right angle (90 degrees) for one of the corners and know the length of at least two sides of the triangle.

Ken K.

I once asked a stupid trigonometry question in front of Bill Engvall and he said, "Here's your sine".

Actually, you don't need right angles. What you need to know is that the side of any triangle divided by the sine of the angle opposite that side is equal to any other side of the same triangle divided by the sine of it's opposite angle (the reciprocal is also valid). The law of sines is thus:

sin a/A=sin b/B=sin c/C where the lower case a, b, and c are all sides of the same triangle, and the upper case A, B, and C are all three angles of the same triangle respectively. If you know a landmark and can get a decent bearing on it relative to north, then you can move in a direction for a given distance (paces) and shoot another bearing and if the two points are relatively far enough apart to generate a significant angular difference, you can then figure out how far you are from the landmark and in which direction. If you know where you want to be (ie base camp) and it's relative location to the landmark, you now have a bearing and rough distance to get there.

Essentially, all you need is two sides and one angle of the triangle, or one side and two angles. You don't need a map except at the very beginning to determine base camp relative to the lanmark. You can also do this if you have two landmarks and no other reference point (including a memorized map location).

Trigonometry is pretty cool. Even if I hadn't been an electronics tech, I would still have found a use for that once boring class.

Here's an excellent link that demonstrates how to use the law of sines, including the deductive arithmetic that starts to make it work with just a compass.

http://www.wisc-online.com/objects/index_tj.asp?objid=TMH2702

If you need more help or haven't gone through a trig class (I highly recommend it), you should contact a math professor and see if you can get in on a class.

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="" />

Absolutely fantastic!!! <img src="/images/graemlins/grin.gif" alt="" /> <img src="/images/graemlins/grin.gif" alt="" /> <img src="/images/graemlins/grin.gif" alt="" />

That is exactly how it gets done. Now if your math skills are properly honed, and you either memorize the sine table or bring a piece of paper with the table printed on it with you, you can do the math without the calculator. If you have a good medium (like snow, sand, or dirt) and a stick, you can draw or sketch the triangulation out without the pad of paper and pencil. It is rough, but it does work. Instead of using miles as a measure, I count paces, which is easier to evaluate, but if you're good at measuring distance travelled, then miles will certainly work.

Once you have your location figured out relative to the landmark, if you know the location of any other point relative to the landmark, you can calculate the bearing and distance to that location, all without a map in hand, using just a compass. Isn't that just handy as all getup?

I'll admit, the one thing I have to have (besides the compass, of course) is the sine table with me. I've memorized it and forgot it so many times my mind is no longer reliable enough to trust. Did I mention I also wear a calculator watch as my EDC everywhere I go? <img src="/images/graemlins/smirk.gif" alt="" />

Have you tried this with a slide rule because I EDC a pilot watch with the slide rule bezel.
I wouldnt have the foggiest idea how to do sine on it though.

Wow, I haven't played with a slide rule in 25 years. I can't recall what all you can do with a slide rule, but I just bet there's a way to calc the sine of an angle on one.

http://www.sphere.bc.ca/test/howto.html

More than I ever wanted to know about slide rules. Some can do multiplications (and divisions?) of sines.

Ouch! Even used slide rules can be pricey. I think I'd stick with the Texas Instruments TI-36X Solar Scientific calculator for about $20.

Being a Palm PDA user, I wish they'd make an AA or AAA battery-powered Palm, but not anymore. I guess the brainy chips use too much power for the alkalines. They all use rechargables and the solar chargers are just too pricey for me.

I went there as well as quite a few other sites.
Unfortunately my watch has the standard log scales and the time scale with no room to fit in the sine scales described on this site.
I made the mistake of quickly loging in here before I went to bed last night and after reading Ben's sine thread and googling, over 2 hours later I still hadnt found what I was looking for. I did read a lot about slide rules - if only I could remember half of it <img src="/images/graemlins/smile.gif" alt="" />

I cant beleive they are charging those amounts and that people are buying them.

I never liked the Texas Instruments sci calcs, I went for the Sharp el-5 series because I found them easier to use.
Ive always got a Sieko Pilot watch on my wrist and mostly carry an iPaq in my pocket.
You get some funny looks when you start calculating with the watch, especially from the 10 to 15 year old age group.

Your pilot watch just has the C and D scales on it, no trig scales. That means that unless you've memorized the sines of some reference angles, you're stuck. That's a pity, because the slide rule would be a great, reliable, fast way to do that calculation. Chained multiplications and divisions, with no addition or subtraction, is where they shine.

There is a Japanese company which makes pocket-sized circular scientific slide rules which would be great for that use, and they're quite reasonably priced. Their number 270 (see here) would do the job, and it's only about $20. Unfortunately, they want 10,000 yen to ship to the U.S. That's about $100, so I haven't bought any of their $20 slide rules yet. Maybe some day I'll stop in Japan and find one in an airport gift shop.

> Essentially, all you need is two sides and one angle of the triangle, or one side and two angles.

Although you may need the cosine rule rather than the sine rule. The cosine rule is more complicated: it says: sqr(a) = sqr(b) + sqr(c) - 2 bc cos(A).

And I think you need the cosine rule to get home. The sine rule will give you your distance to the landmark. So you can draw a triangle CLH where C is your current location, L is the landmark and H is home, and you know the distance CL and LH and the angle at L. But the sine rule can't help you because the angle you know isn't opposite to any of the sides you know. You have to use cosine rule instead. Which means doing squares, square roots, additions and subtractions as well as memorising the cosine table (or knowing how to derive it from the sine table).

Alternative, you can draw a scale drawing. If you have enough information to do the maths you have enough to draw the triangles, and then you can measure the answer with a ruler.

You guys keep teaching me fun new stuff.

Thanks!

OK, so continuing my scenario where we have points:
L=Landmark
A=Current location #1
B=Current Location #2
H=Home

Originally I had taken a bearing of 332 from A to L, then hiked on a bearing of 33 degrees for 1.5 miles to B. Then I took a bearing of 301 degrees from B to L.

From that information, using the Law of Sines I was able to figure out that the defined triangle is:

Point--Angle--OppSideLength
L--31 degrees--1.5 miles
A--61 degrees--2.55 miles
B--88 degrees--2.91 miles

Now the new stuff using the Law of Cosines:
Suppose I know that the bearing from L to Home is 262 degrees and the distance from L to H is 2.566 miles. My goal is to determine the bearing and distance from B to H in order to return home from current location #2.

First I determine the angle between sides LB and LH:
180 - (360-301) - [180 - (360-262)] = 39 degrees

As mentioned, there isn't enough info to use the Law of Sines. Now I can use the Law of Cosines:

BH = SQRT[ 2.566^2 + 2.55^2 - 2(2.566)(2.55) Cos(39) ]
BH = 1.708 miles

Then I use the Law of Sines to find the other angles:
1.708/Sin(39) = 2.566/Sin(B)

Angle at B = ArcSin[(2.566/1.708)Sin(39)] = 71 degrees

Angle at H = 180 - 39 - 71 = 70 degrees

The bearing from B to H is 33+88+71 - 180 = 12 degrees

So, to get home from B I need to travel on a bearing of 12 degrees for 1.7 miles.

Drawing pictures helps me a lot with this stuff.

Very cool!! Thanks again for making my brain work a bit more than it usually would on a day off.

Ken K.

Yes my watch has only the C & D scales and also a time scale.
I use it a lot through out the day, I feel naked without it <img src="/images/graemlins/smile.gif" alt="" /> There are some things that slide rules seem to do better than calculators or spreadsheets.
I didnt use them at school, but two of my older brothers did.
One of my friends is a pilot and he has a Jeppesen E6-B and after having a play with it I bought one. Then I bought a smaller pocket sized Jeppesen CR-5 (these are Australian prices so this week you need to line up 73 on the C scale with 10 on the D scale to convert into US dollars <img src="/images/graemlins/smile.gif" alt="" /> )
I also picked up an old Aristo and a Sun Hemma normal type slide rules from a second hand shop for a few dollars each. Both of these have the Trig scales.
Both the Jeppesen's have a Wind triangle calculator on the back side, so I guess with a bit of experimenting you could work out how to do it on the back.

> So, to get home from B I need to travel on a bearing of 12 degrees for 1.7 miles.

Looks good to me. Did you figure it all out just from being told the cosine formula? Kudos if so.

The short version:

1. Took a bearing on Landmark from postion A
2. Walked from A to B
3. Took another bearning on Landmark from position B
4. Use Law of Sines to determine all angles and side lengths for LAB triangle.
5. Using info from #4, bearing & distance from Landmark to Home, AND Law of Cosines to determine distance from B to Home.
6. Use Law of Sines to find all angles & sides for LBH triangle. This gives bearing from B to Home.

This couldn't have been done without both laws.

I honestly can't imagine doing this in the field without a decent piece of paper and a scientific calculator.