# Sine, Cosine and Tangent Trig Help

1. Sep 24, 2007

### Inkcoder

[SOLVED]Sine, Cosine and Tangent Trig Help

I'm in 10th grade, I was just doing my homework when this dawned on me:

How would one find Sine, Cosine and Tangent without a calculator. So if I am stuck in the desert with a stick I could find the Cosine of 73º by stick and sand method.

Regards,
Austin

Last edited: Sep 24, 2007
2. Sep 24, 2007

### robphy

3. Sep 24, 2007

### dynamicsolo

You can start by finding the values for the fundamental angles (0º, 30º, 45º, 60º, and 90º). Do you have the "angle-addition" formulas? That will get you to all the multiples of 15º. You can get the "double-angle" formulas from the "angle-addition" formulas (or infer what the "addition" formulas ought to be if you only remember the "double-angle" formulas). You can get to the "half-angle" formulas from those, which should get you to multiples of 7.5º.

These will all give you "mathematically exact" results, which means here numbers with radicals in them. Are you wanting approximate decimal values? That'll take some time, extracting the square roots...

If you use enough geometry, you can probably fill in some more values by looking at regular polygons, working out horizontal and vertical distances of the vertices and relating them to the angles in the polygons. (I'm assuming you're stranded for a while and are getting really bored...)

To get cos 73º is probably going to require some sort of approximation approach. With enough values filled in, you can probably look at how the trig. functions behave and work out formulas for estimating intermediate values. (When you get to things like Taylor series in calculus, you can get more broadly applicable formulas.)

This is an issue of some interest, because the earliest tables of trigonometric values were worked out entirely without calculation devices and they predate calculus. It's always good to know something about how you can calculate, or at least estimate, values of functions and answers to problems without resorting to calculators or software. (Sometimes it's the only way you have of judging if the result from a calculation or computer program is credible...)

4. Sep 24, 2007

### dynamicsolo

Just thought of something else that'll work if you want this *particular* number. You know that cos 73º = sin 17º (17º being the complimentary angle). Have you learned the "small-angle" approximation? For (theta) < about 0.3 radians, it's pretty nearly right that (theta) in radians is approximately sin(theta) and tan(theta); this becomes more and more precisely correct as theta approaches zero.

Seventeen degrees is about 17/57 radian, which will be roughly 0.30 radians (divide it long-hand -- you only want two decimal places anyway -- or consider that
17/51 = 0.3333... and 17/68 = 0.25 or just note that 3 x 57 = 171). Therefore
sin 17º = cos 73º = about 0.30 .

The calculator says cos 73º = 0.29237... Hah! In your face, Casio fx-115MS!

5. Sep 24, 2007

### Inkcoder

I think all were worthy answers but for a more precise and working with all numbers, I am going to have to give this one too Robphy. Dynamicsolo thank you but that method seemed to only really work with multiples of 5 and 10, and seemed a bit iffy. Your other response got me on to some good thinking, interesting but seems to only work with specific numbers..

Best Regards,
Austin

6. Sep 24, 2007

### dynamicsolo

How precise a result are you looking for? A "desert island" problem generally implies a low-precision or even "ballpark" value (involving what people used to call a "back-of-the-envelope" calculation).

If you're looking for higher-precision values for the trig. functions for all values of theta, the methods will necessarily be calculation-intensive -- hardly a "stick-and-sand" approach.

7. Sep 24, 2007

### Inkcoder

I think I just tried to dramatically stress the idea that there was no calculator. I just wanted an accurate way to do with pen and a paper just for my own knowledge.

-Austin

8. Sep 24, 2007

### dynamicsolo

That was my understanding: I was describing how people developed the values for these functions when they only had paper and pen and a lot of time... The amount of time required to increase the level of precision by another decimal place generally rises by equal factors.