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Methods of Computing Trig Functions

Date: 09/07/99 at 00:04:44
From: A reader writes
Subject: Solving unknown angle without using table

Hello,

I'm trying to understand something. The book says "In right triangle 
ABC, with A as the right angle: tan(C) = c/b"

To find the angle C if I know the lengths of 2 sides (b = 3, c = 4), I 
can use tan(C) = c/b = 4/3 = 1.33.

Without using the tables at the back of the book, what would I do next 
to figure out the actual angle of C when I know tan(C) = 1.33?

Thank you.

Date: 09/07/99 at 08:54:35
From: Doctor Rick
Subject: Re: Solving unknown angle without using table

Hi, David.

I'm not sure exactly what your problem is, but I think you're 
thinking,

"What good does it do to define this thing called a tangent, if all I 
can do with it is look up the answer in a table? I haven't learned 
anything if I have to look up the answer every time!"

Another way to put it: "So how did they come up with the numbers in 
that table, anyway? And why don't they teach us that?"

The fact is, it's not easy to make those tables; in centuries past, 
people devoted a good part of their lives to calculating the numbers 
in the table by hand. And now that we have computers, the methods used 
to compute trig functions (in computers or scientific calculators) 
make use of things you won't learn for a while yet. 

But it's good to learn enough about these methods to appreciate the 
fact that you have tables and calculators. So here is something I 
wrote for another student who wondered how one can calculate a trig 
function by hand.

I hope it helps you a little.

========

Before we had calculators, we either looked up the answer in a book of 
tables (I still have mine) or we used a special scale on a slide rule, 
which is essentially a table on a stick.

That begs the question: How were those tables calculated? I know of 
three methods. I must admit I have never used any of them! A few 
people dedicated their lives to doing the calculations and putting the 
results in tables, and the rest of us bought the tables so we could 
spend our lives doing other things.

METHOD 1:

The first "trig table" was produced by Ptolemy of Alexandria, Egypt, 
in the second century AD. It was part of a book called the Almagest, a 
name given to an Arabic translation, from its Greek nickname "the 
greatest." This book was used by astronomers for 1000 years.

To construct the tables in the Almagest, Ptolemy started with some 
angles that can be calculated using geometry. He could have used 30 
degrees (whose sine is 1/2), but he instead chose to use 36 degrees, 
whose sine can be found from the construction of a regular pentagon.

Then Ptolemy used the equivalent of our half-angle formulas in 
trigonometry. He repeatedly divided the angle in half until he got a 
figure for the sine of 1/4 degree. Finally, he built up the sines and 
cosines of all the angles in steps of 1/4 degree by using the 
equivalent of our trigonometric sum and difference formulas, which he 
came up with.

METHOD 2:

If you want to compute the sine of an angle without computing a whole 
table full of angles, you can use the second method: a formula called 
the Taylor series expansion of sin(x). You won't learn how to derive 
this formula until you take calculus, but it isn't hard to write down:

                    3     5     7          15    17
                   x     x     x          x     x
     sin(x) = x - --- + --- - --- + ... - --- + --- ...
                   3!    5!    7!         15!   17!

where, for instance, 5! (5 factorial) is 5 * 4 * 3 * 2. The series 
goes on forever, but if x is between 0 and pi/2 (and that's really all 
you need), this many terms (9 of them) gives an answer as good as most 
calculators.

There are Taylor series for cos(x) and arctan(x), also:

                    2     4     6          14    16
                   x     x     x          x     x
     cos(x) = 1 - --- + --- - --- + ... - --- + --- ...
                   2!    4!    6!         14!   16!

                       3     5     7
                      x     x     x
     arctan(x) = x - --- + --- - --- + ..., where |x| < 1
                      3     5     7

If |x| > 1, compute arctan(x) = pi/2 - arctan(1/x). The arctangent 
formula, unfortunately, converges VERY slowly near x = 1 (you'd need a 
huge number of terms), so it's not very useful as it stands.

The tangent, of course, can be calculated from the sine and cosine. 
The inverse sine and cosine can be computed from the inverse tangent:

     arcsin(x) = arctan(x/sqrt(1-x^2))
     arccos(x) = pi/2 - arctan(x/sqrt(1-x^2))

METHOD 3:

There is a Web page that explains in quite a bit of detail how 
calculators themselves calculate sine and cosine. There is a way 
that takes less calculation than the Taylor series. It's called the 
CORDIC algorithm, and it requires computing about 100 numbers and 
storing them in the calculator's memory - something it can do, but 
you and I wouldn't like it so much. Also this method is easier when 
you work with binary numbers, which calculators do but you and I 
don't. So for people, the Taylor series method would probably be 
better, but for calculators the CORDIC method is a snap.

Here is the Web page. It starts out explaining the Taylor series 
method, before presenting the CORDIC method.

How Does Your Calculator Work? (Jeff Morgan)
What I haven't seen yet is how computers calculate the inverse 
tangent.

So those are the three methods I know of:

- Ptolemy's method of building a table using half-angle and sum and 
  difference formulas;

- the Taylor series method to calculate one sine or cosine with a 
  nice formula;

- the CORDIC method that calculators use.

I still prefer a calculator or a table, but it is satisfying to know 
how I COULD do it myself if I had to.