I am giving a speech tomorrow, and I haven’t written it yet. I was going to make a point about how a small change in angle can put you on a very different path. Then I realized that I have forgotten how to calculate that. If I were to turn away from my intended path by, say, three degrees, how far off-course would I be after one mile? How would I go about figuring that out?

a full circle is 360 degrees. it’s also the entire circumference. the radius of this hypothetical circle in your scenario is 1 mile.

you’re 3 degrees off… i’ll leave the rest of the reasoning up to you since it’s for hw.

You need the sine function. If you travel x miles and are off course by y degrees, you will be off your path by x*sin(y). If x=1 mile and y=3 degrees, the deviation is ~.052 miles, or 276 feet.

That’s not that impressive, so how about the following: If you decide to walk from Los Angeles to New York but are off by 3 degrees, you’d end up in Baltimore, and who really wants to visit Baltimore?

See formula for chord length.

If I’m remembering things correctly, a reasonable approximation would be the sine of the angle. The sine of 3 degrees is about 0.0523, so after a mile you’d be about 1/20th of a mile off, or 260 feet.

All of this was done in my head, except for looking up the sine. So it may not be perfect, but it should be close enough for rhetoric. As long as you’re not giving your speech to a bunch of trigonometry enthusiasts.

Worse would be your spherical trigonometry maniacs, who would want you to deduct a fraction of an inch on account of curvature of the earth.

I think it is going to be 2x*sin(y/2), which is not the same, surely.

I am assuming the OP means that the destination is one mile from the start. In that case, the start point, destination point and actual point arrived at form an *isoceles* triangle with an apex angle of 3 degrees. In order to use the sine function you need to drop a perpendicular from the apex, which will bisect the angle there, and also bisect the base in a right angle. This gives you a right angled triangle with one mile for your hypotenuse and half the base for your opposite side, from which you can calculate half the base using the sine function (sine=opposite/hypotenuse). The distance you will be from your destination point is the **full** base of the isoceles triangle.

Note that this answer is not actually correct. There’s a difference between the distance between the two points and the length of the circular arc.

With an angle of only a few degrees, and with the precision requirements being merely those of an illustrative example in a business presentation, I think we’re quite justified in applying the small angle approximation.

If you’re traveling on the surface of a sphere, and you deviate from your course by just a fraction of a degree, then after 20,000 km it won’t make any difference.

Thanks for all the helpful responses! I’m beginning to wonder whether I ever learned this stuff in the first place. I have vague memories of using the “sine” button on a calculator, but it’s been a while.

**CJJ***, I hope you don’t mind if I steal your L.A./New York/Baltimore illustration. I’d come up with one of my own, but I don’t think I can learn trigonometry, write a decent speech, and find my dress shoes all in one night.

This is correct. The function involved, twice the sine of half the angle, is called the chord and is actually older than the sine function. The reason the sine and cosine took over, is the addition formulas such as sin(x+y) = (sin x)*(cos y) + (cos x)*(sin y). But for an angle of 3 degrees the difference is slight. Using sine, you get 276.33 feet and using the chord, you get 276.43 feet.

There is a rule of thumb called the “one-in-sixty rule” used in navigation. It says that a one degree error in your direction will put you one mile off your track for every sixty miles you travel.

For example: if you find yourself 2 miles off your track after travelling 30 miles, you infer that your heading is wrong by 4 degrees, and if you have a further 60 miles to go, you can calculate that you will need to correct your heading by 4+2= 6 degrees to track directly to your destination.

While this is only an approximation it works well for small angles and doesn’t require the use of any trigonometry, sine tables or calculator.