Calculators and Trig Functions

Date: 12/03/96 at 20:35:54
From: Jared Kuhn
Subject: Trigonometric Functions

I have read through your archives and found that the only way you
present to solve a trigonometric function, for example, sin (.78539), 
is through a complicated formula:
 
sin (.78539) = .78539 - .78539^3/3! + .78539^5/5! - .78359^7/7! ... 

Is there no simpler way to do this?  It doesn't seem logical to me 
that a calculator could do this as fast as it does, especially without 
a radian measure to start with.  I understand the principle of the 
unit circle, and was wondering if there was a way (possibly outside 
the realm of trigonometry into calculus) to determine the y value on a 
graph of the arc subtended by the angle theta. This gives the value 
of sine as 1/y (1 divided by the height arc in relation to the 
x-axis). I have asked teachers at my school, but none could make it 
clear to me. Perhaps you can help. I only ask about sine because,
after reading some of the other letters here, I realized that the 
other functions can be derived from sine.

Date: 12/04/96 at 12:29:04
From: Doctor Jerry
Subject: Re: Trigonometric Functions

Hi Jared Kuhn,

I recently went to a mathematics conference at which Prof. Bruce 
Edwards gave a paper titled "How Do Calculators Calculate."  
Here are the first sentences of his paper:

   "How do calculators quickly produce the values of transcendental 
    functions?  For instance, how does the TI-85 calculate tan 1.5?  
    The calculator does not use series or polynomial approximations, 
    but rather the so-called CORDIC method."

I'll try to give a short description of the CORDIC method, but you may 
want to refer to the paper by Charles W. Schelin, "Calculator Function 
Approximation", _American Mathematical Monthly_, 1983, pp 317-325.

To calculate the sine and cosine of a number t, we first need to 
clarify some notation.  We use x_{k} to mean x sub k; x_{k+1} would 
mean x sub k+1; and so on.  Also, x^k means x to the power k; x^{k+1} 
means x to the power k+1.  For -pi/2 <= t <= pi/2, let :

x_{k+1} = x_k - d_k*y_k*2^{-k}

y_{k+1} = y_k + d_k*x_k*2^{-k}

z_{k+1} = z_k - d_k*arctan(2^{-k}).

The constants arctan (2^{-k}) are stored in the calculator.  I don't 
know how many of these are stored, but judging from Edwards' comments, 
it might be something like e_k = arctan(2^{-k}) for k = 0,1,2,...,50.  
In any case, if you want more accuracy, you need to store more of 
these constants.  We also store the product K = cos(e_0)*cos(e_1)*...*cos(e_{46}).  
The choice of the number of factors in this product depends upon the 
accuracy wanted.  The constants d_k are defined by d_k = sgn(z_k).  
The sgn function gives +1 for any number greater than or equal to 0 
and -1 for any number less than 0.

To do the calculation, take x_0 = 1/K, y_0 = 0, and z_0 = t. 

Calculate x_1, y_1, and z_1 from the above formulas.  Then calculate 
x_2, y_2, and z_2.  Etc.

Note that only addition and multiplication are required.

With t = 1, after a while, one gets

x_{47} = 0.54030230586814,

y_{47} = 0.84147098480790,

z_{47} = 0.00000000000000.

The x and y values are the cosine and sine of 1 radian.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/