How would you calculate the sin,cos and tan of an angle without using a calculator?
Yes, you could resort to thisThe OP asked about calculating the values, not looking them up in a table or a book.
but he is askingOne way they can be approximated is based on Taylor series.
Hiring Rain Man to run mental calculations?without using a calculator?
Heh, there is always old school (which I've done). Tables, lots and logs of tables, with interpolation.
Believe it or not there was math before calculators or computers. It is probably no coincidence that early computers were used to... create more tables for artillery in the army.
Interpolation seems quite accurate. I try
Mental calculations are good because it does us good. Using calc/comp technology is not good, there a luxury. Nothing requires a calculator. Faraday, Tesla, Eastwing etc, did they need one? NOYes, you could resort to this
but he is asking
Hiring Rain Man to run mental calculations?
Last I checked interpolation is math, using tables. Tables were the way it was done until the late 1970's, they were a part of the math (that or the slide rule). I am of the age that bridges that gap. It was pretty much the way it was done for many centuries previous. The only people deriving sine function numbers were the table makers, every one else used the tables.The OP asked about calculating the values, not looking them up in a table or a book. One way they can be approximated is based on Taylor series. See http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions
See also the section on computation: http://en.wikipedia.org/wiki/Trigonometric_functions#Computation
You can do far better than that, see for example:Depending on the accuracy you need, a truncated Taylor or Maclaurin series can be quick and easy. It's really just arithmetic.
Maclaurin (and Taylor) series are generally inefficient for numerical approximation as they give very low error around the expansion point and very high errors at the range limits.I still think a truncated Maclaurin series is the best way to go, especially for trig functions.
In maths there are many equivalent definitions for sine, in analysis it is usually defined by:Sin and cosine function ARE based on the ratio's of the sides of a right triangle.
How they are used in other math problems(even phase angle calculation) does not change that basic fact.
Agreed. I was under the impression that the OP wanted to calculate on specific value. Still, a 4 term Maclaurin series will give you decent results from -∏ to +∏ for sine or cosine. all depends on your accuracy requirements.Maclaurin (and Taylor) series are generally inefficient for numerical approximation as they give very low error around the expansion point and very high errors at the range limits.
But it becomes a matter of effort. The method you linked to is good, but requires the use of tables (a(n) values) and calculations. If you are just interested in a single value, not a viable range, this is more work.The series in Abramowitz and Stegun that I linked to were not truncated Maclaurin series. For example, if you look at the approximation there for sin(x) using only three terms (4.3.96), the maximum error over the range is a tenth of the error for the truncated Maclaurin series at x=pi/2 (Maclauren series with the same number of terms).
I'll look them up. Thanks.If you are interested, one way better approximations can be generated is by truncated Tchebycheff expansions, these are not optimal but they can be pretty good.