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Function Value Calculations
Name: Miriam T.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: 11/20/2004
Question:
The actual value of the sine of an angle is given a
value, on my calculator, to 8 decimal places. How is this value obtained?
(surely not by accurately drawing a right-angle triangle and measuring
the ratio of opposite side to hypotenuse! -or is it?
Replies:
Miriam,
I remember writing an old program that would "estimate" the value of sin of
(x) using Taylor series expansions. Or maybe they were called it the
McLaurin (that is when the function is evaluated @ x = 0). I think it is
called the Taylor's Series expansion at all other non - zero values of 'x'.
I may be wrong but it is possible that your calculator could do a 9th or 11th
OR EVEN a 13th order expansion of the transcendental functions in order to
get "reasonable" approximations (6-8 digit accuracy).
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + (x^9)/9! ...
Note the following:
3! is also called "THREE FACTORIAL"
3! = 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
Note the sign changes: + - + - + - + - ......etc......
As an exercise...open up Microsoft Excel and try comparing sin(x) {the
built in function} and write the formula for an 11th order Taylor
approximation for Excel's INTRINSIC function SIN(X). You'd be surprised how
close they are.
I hope this helps some.
Regards,
Darin Wagner)
Miriam,
Thanks for your question. If you think about the graph of sin(theta) with
the angle theta plotted on the x-axis and the value of sin(theta) on the
y-axis, the graph is a smooth "wavy" curve that oscillates between -1 and
+1. Therefore, the value of the sine of an angle covers EVERY value between
-1 and +1 -- even values with an infinite number of decimal points (your
calculator only shows 8 because that is the maximum number of digits that
can be displayed on the calculator's register). This requires denoting
finer and finer measures of the angle -- whether in degrees (sin(4.5
degrees) is different than sin(4.55 degrees) or radians (sin(3.14) is
different than sin(3.14159)).
Regards,
Todd Clark, Office of Science
U.S. Department of Energy
Very perceptive. This is something that you can introduce to your
students. There is a branch of mathematics call infinite series which
deals with algebraic expressions for various functions. One text is
"Theory and Application of Infinite Series" by Konrad Knopp. There are
many others that you can find if you do a Google search on the term
"infinite series" (one site is:
http://www.math.unl.edu/~webnotes/classes/class46/class46.htm#psexpfcn
Without proof:
The formula for sin(x) = x/1! - x^3/3! + x^5/5! - ... +(-1)^k *
x^(2k+1)/(2k+1)! ...
And for cos(x) = x/1! - x^2/2! + x^4/4! - ... + (-1)^k * x^2k/2k! + ...
Recall k! is " k factorial" k! = 1*2*3*4*...*k
Since the right hand side of the " = " sign is an algebraic expression, one
only needs to add as many terms to the series as necessary for the desired
degree of precision for the trig functions. There are other series
expressions of these functions that may converge to the desired precision
with fewer terms, so I don't know that these are the particular series used,
but it gives you the idea.
Vince Calder
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